Michael Huemer and the Eternality of the Past

Michael Huemer is an interesting guy. I don't have hard data on this, but I'm fairly confident that he's the only philosopher working today who simultaneously advocates veganism, anarcho-capitalism, agnosticism, and substance dualism. As such, it was not terribly surprising to find that he also believes in reincarnation, having defended the idea in a (very interesting) 2019 paper in Nous. The gist of Huemer's argument is as follows:

If you were to live at most once in the history of the universe, we cannot justify assigning any probability greater than zero, initially, to your living at the present time, given an infinite number of past centuries in which you could have been born instead. The best explanation for your otherwise surprising present existence is that you are repeatable, so that your living at an earlier time would not have prevented you from living now – in which case, your present life also will not prevent you from living any number of future lives.

Now, it occurs to me that this argument could be used, not only to argue for reincarnation, but also to argue against the eternality of the past. Indeed, one might think that it is a somewhat stronger argument for the latter, seeing as most people will find reincarnation extraordinarily counter-intuitive, whereas the idea that the universe had a beginning is vastly less so. This is especially true seeing as many philosophers have a view of personal identity on which reincarnation is metaphysically impossible, whereas almost nobody is committed to any theses requiring the eternality of the past. As such, when confronted with an argument which says that either the past is finite, or reincarnation is actual, it is plausible that most philosophers will opt for the former.