I guess it all depends on how we define 'soul'. Since Buddha is unlikely to give any coherent definition...
I'm working under the assumption that he's attempting to prove the Buddhist formulation of reincarnation. Yes there's a kind of supernatural goo that facilitates reincarnation there, but it's not discrete animism as you'd expect in, say, a Christian formulation of resurrection. Buddha the Claimant is trying to avoid the empirical entanglements of having to prove the existence of this goo -- which, naturally, is a hidden premise to his claim -- by denying any sort of mechanism and pointing out that Buddhism rejects individualized animism. Yes, we know Buddhism rejects animism, and none of our criticism is based on animism. So it's a straw man.
And his other tap dance today (to sidestep the problem of mechanism) is insisting that reincarnation must be like quantum mechanics, in that we prove only the predictions and not the underlying mechanism. He claims we shouldn't have to prove any underlying mechanism for reincarnation either. The main reason that's wrong is the conversion problem; he wrongly assumes quantum mechanics and his proof for reincarnation are categorically equivalent. They aren't. But it's fine with me if he wants to point to all the problems of empiricism with quantum mechanics and imply that his proof for reincarnation is similar. It just means that his proof for reincarnation has the same problems with empiricism.
This is what's really silly. He promises an empirical proof. Then when his critics point out that it's not empirically valid, he says he doesn't believe in empiricism, or that the nature of what he's studying defies empirical expostulation. I just laugh when he says his critics don't understand proof. Buddha can't seem to see all the times he shoots himself in the foot with his own rejoinders.
Quantum mechanics is a set of deductive models that follow from earlier work. The models are probabilistic because probability is how we model things whose mechanisms we don't understand or which are too complicated to model practically in all their gory detail. The model does not propose a mechanism, which is not the same thing as asserting there isn't one. Having deduced what must follow from earlier work, we can certainly make observations that test what should then follow further. Entanglement, for example, is something that would follow from QM, and we follow the deductive path from entanglement until we arrive at some deduction we can observe. We don't know what mechanism makes entanglement work, but we can confirm that it happens. This is what makes the inquiry empirical instead of inductive.
But reincarnation doesn't follow deductively from anything. It's not something that
must be true as a consequent of something we can observe. In contrast, It's something hypothesized to explain an observation. Hence Buddha the Claimant's reasoning is the converse of the reasoning in quantum mechanics. But given enough detail in the hypothesis, we can deduce what must follow from any of those details. That would be "mechanism." And we follow the line of deduction until we get to something that can be observed. The observation of the consequents of the proposed mechanism confirm the mechanism. Ideally we formulate the observation so that the absence of it falsifies the mechanism, and with it the hypothesis.
For example, you rightly assert that if reincarnation occurs, and that it involves the persistence of memory from one physical incarnation to another, then there must exist an entity that has the property of retaining memory and which doesn't depend on a functioning human organism. That follows deductively from the hypothesis. What then might follow deductively from that? Keep going until you find something that is susceptible to observation and you have your empiricism according to the hypothetico-deductive method.
Agreed, as with almost everything he's said he's not really addressing the criticisms in any useful way and insists on tossing out counters to arguments nobody is making or side notes nobody but he finds relevant.
Correct. He's trying to convey the illusion of participating in a meaningful review of his claim without having to address its flaws. I feel that claimants do this because it has worked for them in the past, either because they insulated themselves from any criticism or because they presented their claims only to sympathetic audiences. The denouement of this is frequently restating the purpose as simply collecting feedback. The proof fails, but he is vindicated because he acquitted himself well in the debate. This is an ego reinforcement exercise.