It’s important to specify what is meant by ‘self.’ In English-speaking Buddhist discourse, this word easily gets thrown around in all sorts of ways. But it often refers to very different kinds of things.
In the EBTs, ‘attā’ is also used in a variety of ways. But in the main philosophical sense, it means a permanent or unconditioned substance, essence, or experience that is independent of other things. If this is understood, a lot of confusion can be dispelled.
For example, the idea that a permanent consciousness remains but that it is “not personal” or owned by anyone, is still positing an attā as far as the teachings of the early texts are concerned. Because ‘attā’ in this context doesn’t mean the concept or notion of a sense of self/‘ego,’ but means the unconditioned reality/essence mentioned above. ‘Anattā’ is generally the crowning point derived from anicca, dukkha, and paticcasamuppāda. Even in the Anattalakkhana Sutta, the argument goes that if form were attā we would be able to exert control over how it is at will and it would be only sukha. But because form is dependently originated and dukkha, it is anattā.
That doesn’t mean that form can exist as an eternal “impersonal” thing. It means that it is dependently originated, dukkha, and impermanent, and so therefore it is liable to cease and should not to be taken up or appropriated with a sense of self. In other words, ‘ego/sense of self’ is the psychological effect/outcome that comes from having an ontological/metaphysical belief or notion of a permanent, independent thing (attā). Or independent substance in general, including if it is impermanent (i.e. the view of annihilationists, that the self is destroyed).
So, in terms of the existence of the khandhas vs. the attā. We do experience, for example, form in existence. So we could say that, obviously, form exists. ‘Exist’ here is what others would call ‘conventionally exists,’ in that it does not have a separate existing substance to form, but it too is a temporary, dependently arisen and conditional experience. But we can designate an actual experience of ‘form’ as a classification or way of categorizing experience in samsāra.
Nonetheless, when we break it down, ‘form’ does not exist as an independent substance. It too is a conditional experience. But the suttas never claim otherwise, so claiming that the temporary, conditional arising of ‘form’ exists is no problem. It’s claiming that form is an independent substance that is a problem, because this amounts to an attā which leads to a sense of self and craving/upādāna which leads to continued existence and rebirth in samsāra.
On the other hand, we cannot designate an actual experience of ‘attā.’ Because it is impossible to experience a permanent, unconditioned essence. The very nature of conscious experience is that it is dependently arisen and temporary, so to experience a permanent ‘object’ cannot happen. This means that people misperceive and miscosntrue what are conditional, temporary experiences by believing that they are experiencing an essence/substance/core unconditioned reality. But that experience is not actually that idea; the idea is not actually the ‘attā,’ it’s just the notion of an attā.
In other words, because ‘attā’ in the early discourses would refer to something that is permanent and unconditioned, we cannot talk of the temporary conditional arising of the ‘attā’ in experience. That is a contradiction in terms. And so the discourses would not say that obviously ‘attā’ exists.
This means that the idea or concept can be said to obviously exist. And this is clearly done in the early discourses. Even the notion of a personal individual, like arahants saying ‘I/mine’ or saying ‘The Buddha saw X’ etc. is recognized as an existing convention. But the discourses would not say that there obviously exists an ‘attā,’ because even in conventional terms no such thing is experienced. Again, only the idea of such an experience is conceived, but the actual experience cannot be categorized as the attā.
As an aside, the notion that we do temporarily experience permanently existing phenomena, i.e. we can come into contact with a permanent substance/essence, was proposed by the Sarvāstivādins. There, the idea is that ‘form’ (rūpa) for example does have a self-existent essence, and we can temporarily come into contact with that experience under certain conditions. The problem is that, as should be clear from above, this ends up being identical to positing an ‘attā’. Because it is positing that we can experience unconditioned substances, and that those unconditioned substances do exist. Notice that they still rejected the idea of a self, and of course argued for ‘anattā.’ But in the language of the early discourses, these substances would still be an ‘attā,’ and to say we can experience them is just forming a concept about temporary, dependently arisen things that leads to construing of substances or knowledge outside of the aggregates, i.e. a sense of self.
This is why Nagarjuna, for example, argued against the Sarvāstivādin position (which is included in the late Theravāda Abhidhamma, AFAIK). He showed that by positing that ‘impersonal’ permanent substances exist still amounts to positing an ‘attā,’’ even if we say “it’s not self!” Because it’s not about positing the existence of a sense of self in this context, but about the concepts of permanence and things being beyond conditions. This then inevitably leads to or involves an ignorant sense of self, hence the issue.
That’s how I’d grapple to summarize my understanding of some of the conflicts here. It would be a complete misunderstanding of emptiness (as I understand the EBT perspective) to accuse this of meaning arhants exist after death. Denying that things truly cease is denying that we are experiencing the cessation of inherently existent substances, not denying that dependent processes come to an end (which is exactly what emptiness is pointing to).
Emptiness is a term that designates how our experience is not one of essences/substances (i.e. attās), but of dependently arisen phenomena. That is, when the conditions for temporary, dependently arisen experiences (i.e. the five aggregates) are no longer there, those experiences will also cease. But no actual substances or essences can be said or seen to be ceasing (or arising). Just temporary designated conditions. This is how there is no notion of eternalism or annihilationism, because no truly existent substance is annihilated here; dependent experiences/processes just come to an end.