I’m not trying to assert anything about the Buddha’s historicity, in either direction. But in this recurring hot-button topic de jure, I want to bring some context from my exposure to similar discussions in other areas.
I don’t think you meant this to be exhaustive, but I’d add that sources can be downgraded to secondary by factors other than time. An academic book on the Iraq War is a secondary source, and the Wikipedia article on it is a tertiary source. We don’t know exactly when Diogenes Laertius wrote, but even if he met your timeline criteria for proximity to Sextus Empiricus, he probably wouldn’t count as a primary source, since it seems his whole work is a secondary source based on (largely lost) primary sources. I.e. even if he was a contemporary, he likely didn’t base his statement that Sextus was a student of Herodotus on direct personal evidence, but rather based on reading it from a primary source (since he writes about it exactly the same way he does events that must have occurred centuries before his birth).
If we found a contemporary source from, say, China, which was a summary of a compendium on reports of foreign notables and mentioned something like the Buddha, that might move him from pre-history to proto-history, but it wouldn’t be a primary source for his life in the sense you’re discussing.
And even primary sources often come with limitations - for example, Aristophanes and Plato were contemporary to Socrates by the standards you’re suggesting, but both were writing in genres with primary concerns other than historical accuracy. And Xenophon doesn’t really clear things up. The Socratic problem is viewed as being definitively unsolvable. Certain broad details can be derived (e.g. he was executed) but a huge number of important questions are not solvable by historical methods (e.g. did he accept payment for teaching).
Even if we had three written sources from the time of the Buddha, it’s very likely, given the nature of his significance, they would not clearly and consistently describe his life course in the desired level of specificity.
I think there’s an unnecessary addition of hypothesizers deluding themselves into certainty. Scholars of all sorts form hypotheses based on available evidence, and then hope to be able to one day test those hypotheses against new evidence.
Sometimes, these hypotheses can then be tested with great resolution. For example, there’s a US government program that uses historical data to generate hypotheses about where soldiers may have died overseas and had their bodies left behind. They then send teams (of archaeologists &c) to test the hypothesis, locate, identify, and recover those bodies. The people who do this know that sometimes, even with their strongest hypotheses, they may be wrong. But they do the work because it can bring 100% certain closure to these stories.
Sometimes, these hypotheses can’t be tested - for example, there’s no conceivable way to test many hypotheses about Socrates.
More often, it’s somewhere in the middle. For example, there’s no plausible way to ever prove that Snorri Thorfinnsson (a legendary Norse Christian Chieftain) was born in North America. But various evidence has built up to make it more plausible - before discovering L’Anse Aux Meadows, there was less reason to believe the Vinland settlement happened at all, and before specifically finding women’s tools like a loom, there was less reason to believe that women were ever there (a precondition for giving birth). No serious person would say they’re 100% confident he was born there, but a serious scholar would acknowledge that this evidence changes the plausibility of the historicity (quality of actually having happened in the past) of that legendary event.
Recently there was a massive effort in China to revise the chronology of their earliest dynasties. In a rare display of the CCP taking a liberal approach to dissent, there were immediate disagreements about conclusions and admissions of uncertainty. As a whole, the people embarking on this project didn’t end up deluding themselves into some dogmatic certainty.
Primarily but not entirely. In general in academia, interdisciplinary studies are all the rage. At a basic level, the interpretation of historical documents always involves knowledge from other domains, as even someone just working from documents has to bring background knowledge to bear to understand descriptions of events (e.g. records of boat trips rarely explain the principles of buoyancy), but also many historians either bring in resources from other fields or are working as a resource for another field. A historian’s core competencies are in written records and scholarship about written records, but historians tend not to be utterly incompetent at the things which touch on their core. And even in the written record, there is more to be expert in than just the history of a society’s historical era (era with narrative writing about contemporary events from insiders) - historians do also look at pre-history (events that took place before such writing) and proto-history (the period when a culture was written about but didn’t write narratives about themselves).
For example, the pre-contact Amazonian city states are proto-historical (written about by outsiders, not themselves). For a long time, historians considered the descriptions of their size to be exaggerations or fabrications, based on the seeming implausibly of building at such a scale in such an environment, the non-urban findings of later explorers, the obvious economic and social motives for such exaggerations, etc. Further developments from other humanities (for example- the presence of hereditary aristocracy in post-contact non-urban societies) made it gradually more plausible, and then progress in hard sciences (eg LiDAR) helped even more. But, critically, it’s inconceivable to imagine new evidence supporting or harming certain claims, like the claims that baptisms of natives were successfully conducted.
That term is historiography - the study of the writing of history. A study of the History of WWII might include insights from recent excavations, using them to inform historical documents. A study of the historiography might compare how an event was written about over time, but would adhere strictly to what was written.
Another point to be made is that historicity and history do in fact have two overloaded meanings:
- The documented past
- The past
The distinction is totally clear when we talk about the far past. For example, the Permian–Triassic extinction event is a part of earth’s history. Nobody is confused with what I mean by that.
It’s also fairly clear when we talk about recent things that we write about but don’t think of as having a physical representation. For example, on April 25th 1898 America Declared it had been at war since April 21st. That’s time travel. A later event (a bill being signed into law) can’t cause a prior state (war). But we get that in actual events on April 24th there was no declared war, yet in another sense there was.
The tattooing of “Otzi” the “iceman” can be inferred to have happened. It would be mildly confusing to call it a historical event.
Certain details about Egil Skallagrimson can be archaeologically verified (e.g. his big skull) and others examined with “soft” evidence (the poems attributed to him have several markers of authenticity). He’s from a proto-historic culture (he’s said to have fought in battles we have primary sources for, but those sources aren’t from his people). He’s also a legendary werewolf with several attributes and deeds that can’t have really happened in this physical universe as we understand it. It is very confusing when he is called a historical figure (though I’ve heard Icelandic people do this), which is why I believe most Norse specialists prefer the term “Legendary”, which denotes a story with some basis in “history” (things that happened) but not in “History™.” However, even as a legend, the historicity of Egil is not in question, in the sense of having really lived in the real past.
I don’t want to mislead by talking too much about the actual issue at hand here, but for a closer tangent consider the Mahavamsa. It says the Sinhalese came to Sri Lanka lead by a Northeast Indian descended from a lion, who fought Yakshas, etc. The part about the lion definitively rules it out as history. But genetic evidence linking the Sinhalese to Northeast India makes it seem like this might be a legend with some faint trace of “history”. These sorts of stories are at the opposite end of the “legend” spectrum from figures like Egil or other figures from just before the start of a culture’s historical era, and are also called “origin myths”.
The lack of contemporary written sources rules out proving the historicity of the buddha in that sense or on that basis. The open question (which I sincerely am not trying to imply a conclusion about) is: are there any other ways of establishing his historicity (in the sense of being part of the series of events which really happened). A DNA analysis of all the monks & nuns (the sons & daughters of the Sakyans) clearly isn’t going to do it 
Secondary sources, primary sources which allow for inference, and evidence from other fields.
More or less name any European people other than the Greeks. Their “history” will include a time when all we have are “History ™” passed down by outsiders and legends passed down internally. This is because they had neighbors who were obsessed with writing history, and then more or less as soon as a people adopted writing, they had an elite with strong incentives to glorify their ancestors in writing. We can’t say much definitively about these times for these peoples, but historians do speak on them carefully and informatively.
Another example Jayarava brought up is the case of the Polynesians. They did not have written histories, but they had many other features, such as an intricate storytelling and mapping tradition and isolated geography which mean we can lend some credence to the oral tradition about their migration history, even in some striking specific detail (e.g. the ~1200 AD contact event between Polynesian and Andean peoples). As well, I think the relatively intensive and parallelized efforts to write down their oral history gives us, at very least, a better take on its reliability than other indigenous groups.
This is an issue of the genre of texts, in addition to what you’re saying about their age and medium.
I think it’s worth considering that basically all pre-modern history is de-mythologized history, and, in the general imagining, de-spiritualized history. Ordinary people think of Julius Caesar not as high priest, deifier of mercy, and postmortem god, they think of him as a secular ruler who got stabbed on a specific date. The first written account of a book in Scandanavia is a story (written by a close companion of an eye witness) about a bible which an illiterate man could read through miraculous powers. Holy Roman Empress Cunigunde of Luxembourg is an important figure in the history of Christian royal women, with hints here and there about her degree of influence over important decisions. But the clearest narratives about her describe her miraculous powers over fire. Orderic Vitalis, a credible secondary source in many ways, and Ambrose of Normandy, an important primary source about Richard the Lionheart, both credulously discuss Gobelinus the shapeshifting demon which haunted Évreux. George Washington & the Battle of Valley Forge were mythologized by contemporaries, with written (though problematic) accounts of Washington’s intercessory prayer and visionary exchange with the Virgin Mary. The emancipation proclamation was attributed to a vow between Abraham Lincoln and the Christian God. George Bush claimed God told him to end tyranny in Iraq.
The more recent you get, the easier it is to just cut out the myth and religion and retain a coherent narrative. You can safely take a lot of modern invocations of the divine as mere turns of phrase, or limit them to psychological phenomena. In medieval Europe (~1000-1500 CE), it is my sense that current professional historians think the public underestimates the prevalence and seriousness of these psychological phenomena, but they’re still able to strip the mythology off the skeleton and hang a sensible narrative over dates & worldly events. The “Dark Ages” (~500-1000CE) in many places lose this skeleton (or at least some bones - you might have a tibia here and a fibula there). Then the Greek & Roman historical era (~500BC-500CE) has the skeleton once again, but there’s an issue of us not understanding how to psychologize the mythology. In the medieval era, we’re more confident in our understanding of the power of Fragments of the True Cross or the Spear of Destiny or even the incentives in hagiography for the purpose of securing sanctification of the body you controlled and could potentially turn into relics / control access for pilgrims creating relics (by e.g. touching cloth to the bones). We don’t really understand the Eagles, or the Lares, or the Mysteries, or the Genii Gens, or the Genii Loci, etc. etc. in the same way. And then when we go into the Archaic period, or Greek & Roman Pre-History, there’s some figures from late enough pre-history people speculate about the fine details of their de-mythologized historicity (e.g. Hesiod & Homer) and then there’s a deeper past people do try to historicize in broad strokes (e.g. Troy).
This demythologizing historiography itself has an interesting history, which I think is best demonstrated by Snorri Sturluson’s Ynglinga saga (but goes earlier). There, Sturluson, who seemingly believed in the preternatural but not the gods as such, attempted to (by his standards) rationalize Norse myths as echoes of real events, using false etymology to equate the Aesir with Asians, turning Odin into a great Asian chieftain with powerful magic, and Freyr as a chief of ~ Europeans, the ancestor of the Swedish kings. Humans have, for a long time, wanted to figure out how to hold on to both their views of how the world works and their stories from the past.