our perceived history matches the Born rules too well for it to be reasonable that "probabilities are meaningless", so either the universe is OK with measures on infinite sets or it's somehow finite after all

I like this; it is an excellently compact way of putting it.

The OP was careful, it seems, to avoid that issue. (Infinite set agnosticism?)

I don't see this. I am referring particularly to this paragraph:

No, this view of reality claims that your current observer-moment is repeated infinitely many times, and looking forward in time, all possible continuations of (you,now) occur, and furthermore there is no fact of the matter about which one you will experience, because the quantum MW aspect of the multiverse has already demolished our intuitions about anticipated subjective experience.

I agree with you that the Born rules imply meaningful probabilities; but it seems to me that the OP does not believe this, at least in the part I've quoted.

Yes, I understand this, but as noted in my comment above, it appears that the OP is using a different assumption.