Greetings! 
Good to see you 
I apologize in advance: I do not have the energy to make complicated posts with well cited examples about this issue. You, someone else, or I may end up convincing myself later on, but as of right now, I’d prefer not to spend the hours it would take.
That said, this is a major interest of mine nowadays. And what I can say is that Bronkhorst’s ideas on rebirth and karma have been proven heavily wrong by the ingenious Vedic scholar Joanna Jurewicz. She has also demonstrated that rebirth goes back all the way to the RgVeda, and she has demonstrated the importance of it as it evolved through the Brahmanical tradition even post-Buddha.
She has three books on the development of Vedic ideas. One on the Rig Veda, the next on pre-Buddhist texts (Rig Veda, Atharvaveda, Brahmanas, and early Upanisads), and another on post-Buddhist smrti texts. They are all very long and complicated, yet extremely insightful and full of colorful information.
All the way back to the Rig Veda, the idea of a permanent heaven went out of fashion rather quickly. The idea that even in heaven, people relied on food that was earned via ritual and faith, etc. grew early on. Eventually people would be rained back down to be born in their family and build up a space in heaven with ritual again.
I recently read this rather short yet insightful article on the topic:
Food and Immortality in the Veda: A Gastronomic Theology? by Carlos Lopez. You may find it helpful
Another very helpful one that is short is by Joanna Jurewicz: Rebirth eschatology in the Rg Veda. It discusses how this belief worked in small-scale Vedic society as opposed to the evolving ideas of rebirth past the Rig Veda.
A more general overview by Gregory Shushan discusses rebirth and other afterlife conceptions in the Vedas here. This discusses non-rebirth-esque views that existed as well and how they relate or differ. This is probably one of the best overviews on the subject in a very brief, neat, and clear way. It goes through several Vedic passages and consistently shows how ideas of rebirth go back as far as the Rig Veda and are extremely apparent in other Brahmanas, etc. (but it is not all uniform: some may be annihilated, others immortal, others reborn again and again, etc.).
Despite references to immortality and to the world of the immortals in the Rig Veda, there are early indications of a belief in rebirth as the inevitable conclusion to a temporary stay in a heavenly realm (at least for those who escape annihilation), the length of which is determined by one’s ritual conduct on earth.
Gregory Shushan
(This agrees with the article by Carlos Lopez and Jonna Jurewicz on rebirth being found in the Rig Veda, ‘immortality’ not actually being permanent, etc.)
The scholars you quoted as saying that rebirth was virtually non-existent are sorely, sorely mistaken. To be honest, one would almost have to never have read these texts in any depth to say that the idea is just “the soul goes to heaven or hell” as Reat does, for instance. That idea is so clearly false with even a cursory overview of the material. A more detailed analysis of a passage is in the Jurewicz article. I leave an excerpt from her conclusion here related to rebirth:
Let us recapitulate the rebirth eschatology reconstructed on the basis of the Rgvedic evidence: the dead person, properly cremated, was poured as a Somic oblation into the cremation fire. He reached the sun, where he enjoyed the contradictory afterlife state. Then he was sent back by the sun in the form of rain to be reborn among his relatives. This rebirth eschatology has all the features of the rebirth eschatology characteristic of small-scale societies.
Even beyond rebirth, the afterlife is much more complicated than a heaven hell duality or some immortal soul in Abrahamic heaven. There are of course parallels, but as Shushan, Lopez, and Jurewicz (staying with only these three for now) demonstrate, the view must be from scholars who were unfamiliar or before Vedic scholarship had developed to understanding the texts very well.
I may come back later and edit in citations; I apologize again for the lack right now. I just thought it would be good to at least drop Joanna Jurewicz’s name here who is probably one of the world’s leading scholars on this issue if not the leading scholar. Any critical read of the Brāhmanas/early Upanisads will show that their ideas of rebirth and karma are not just out-of-place additions from other ideas; they are extremely consistent and knowledgable with centuries of earlier Vedic tradition and ideas.
Another thing to keep in mind is that the standard description of ‘right view’ (mundane right view) in the suttas (such as in MN 117) is highly general and is not anti-Brahamanical; if anything it utilizes a lot of Brahmanical terminology. The terminology there speaks of ‘this world’ and the ‘other world,’ of sacrifice and giving, of basic ideas of morality leading to rebirth, etc. Although Brahmanism was not a unified institution and indeed many ideas existed, the description of right view is not at all at odds with what many brahmins would have thought. We can also add the fact that in the earlier strata of suttas, the Buddha and (especially contemplative) Brahmins are ideologically much more compatible in terms of various realms of existence, our actions influencing our afterlife, etc. This was discussed in the PhD dissertation Early Buddhism and Its Relation to Brahmanism by Gabriel Ellis (who is or was a poster here as well!). I don’t want to exaggerate the similarities as some in the past have done, but I do think this is worth pointing out. There is a good essay on this from the forum here some may be interested in.
All of that said, I agree that the idea that the Buddha blindly adopted rebirth from some mainstream Brahmanism is mistaken. You make a good point too in pointing out that the ideas of contemplative brahmins such as those found in the Upanisads were not the mainstream for the average person, although the Buddha was likely familiar with these as we see in the suttas. He had a very different idea of rebirth, of the deities, of the process, etc. from many people. In more orthodox Brahmanism, it’s all about ritual (even though moral behavior is also an important factor going back a long time); it all involves pre-scientific views of natural processes and travelling among astral bodies; the evolution of consciousness and craving do not drive it (though ideas about this evolved over time and have earlier roots); the process is not always universal, ‘egalitarian,’ or consistent; there are several factors involved from external agents (family members giving offerings or priests preparing your funeral/rituals and so on); etc. The Buddha brought many unique ideas to the table, but the idea of rebirth itself was not at all unknown to the Vedas or Brahmanical tradition. I also agree that the Buddha was probably a major factor, if not the main contributor, to rebirth gaining such popularity in later Indian philosophy. This seems to be one of the most impactful aspects of Buddhism on Brahmanism and into Hindu systems!
EDIT: Two translations of RgVedic passages on rebirth, discussed by Jurewicz above.
‘Release him to his fathers and again down from them, who, poured into you, travels according to his will. Let him who wears life come to his offspring. Let him join his body, Jātavedas!’
RgVeda 10.16.5
‘O Agni, sow again the one you burnt [before]!’
RgVeda 10.16.13
(Notice the same farming simile for rebirth that the Buddha frequently uses attested all the way back to the RgVeda!)
I will also leave this excerpt from “Fire, Death, and Philosophy” by Joanna Jurewicz:
Many scholars assume rebirth theory was introduced during the time of the early Upaniṣads and came from royal circles that adopted non-Brāhmaṇical concepts and taught the Brahmins about them. … Yet, scholars like Tull (1989), Killingley (1997) have presented evidence for earlier accounts of the rebirth theory. In Jurewicz (2008b, 2010), I have shown that the concept of rebirth is attested already in the ṚV. Cremation is conceived in terms of a sacrifice in which the deceased is a Somic oblation. As during sacrifices performed in his life he reaches the sun where, having drunk Soma from the solar source, he transforms himself into a perfect being called Aṅgiras. It is believed that on the sun he meets his fathers and Yama. The deeds he performed during his life … influence him which is conceived in terms of their union with him (ṚV 10.10.14.8). Then he comes back in the form of rain to be reborn as a member of his own family. … I have also shown that traces of the belief in rebirth are present in earlier thought (AVŚ 11.5.13-14, 15.7.2-5, ŚB 10.4.3.10-11). … We will also see that the context of the expositions of the JB and JUB attesting the rebirth theory is purely Brahminic.
The Jaiminiya Brāhmaṇa contains a detailed description of how rebirth works in various paths and with the Five Fires (very similar to what is described at BU, CU, etc.), and as a brāhmaṇa, this would be the major text for the Vedic śākhā it belonged to, dictating their ritual action and interpretations of the Veda.
In the book’s conclusion, she writes:
As I have shown, it is possible that the belief in rebirth within the range of one’s family or clan was so well entrenched in early Vedic thinking that the composers did not feel necessary to elaborate it more extensively. … This topic is reconsidered again in the Upaniṣads when a new practice is found which leads to the possibility of not being reborn. Throughout the early Veda, the state reached after death was the state which was reached during supernatural cognition performed in one’s life. According to the earlier tradition, attested in the Saṃhitās and the Brāhmaṇas, the range of the supernatural cognition was the borderline sphere between the two aspects of reality, so it was believed that everyone came back to the earth after death. The Upaniṣadic practice allowed the practitioners to reach the unmanifest aspect of reality.
For male brahmins especially—who seem to nearly always be the only people whose afterlife situation is much addressed—having sons was of extreme importance. Having a son was part of gaining immortality, in that they make offerings for the deceased relatives, and it seems another part of this is that they will bear off-spring in the family through which one could be reborn when the time comes. This was a major point of controversy and tension in the Brahmanical tradition as more and more celibate contemplatives came along. Eventually it led to the development of the āśrama system that many know today in modern Hindu circles. This is discussed by Patrick Olivelle in his work ascetics and brahmins for those interested.
Finally, I’d just add that in the Śatapatha Brāhmaṇa (as is a continuation of earlier Vedic thought and which continues later on as well), reality must manifest itself in constant death and killing to stay alive. It kills itself to live and be born a-new, thus being the killer and the killed. Even Prajāpati, the Creator, dies and is resurrected — what to say of mortal humans who are supposed to repeat his actions in ritual? I would argue that the model of reality presented in the ŚB nearly necessitates a belief in some kind of rebirth—and this is precisely what we see elaborated in the BU section of it in a way very similar to JB, JUB, CU, etc. It seems to me that it was likely taken for granted by many brahmins that reality was cyclical, like a sacrifice with oblations and heat (eater and eaten) giving birth to one another, and that the afterlife of man followed the same pattern. As Prof. Jurewicz says, the Upaniṣads are where the ideas of escaping and transcending this come up; it does not seem to have been perceived of as suffering by the majority at the time. In fact, it was a manifestation of reality, immortality, ritual, winning worlds/space (lóka; something mentioned by name in the suttas and the Brahmanical texts), etc.
From Obeyesekere 2002 on small-scale rebirth eschatologies and their evolution. This is what Jurewicz sees evidence of in the Rig Veda.
That’s my spiel! I best not add more info for now.
Mettā!