Does anyone know, are there versions of philosophy-of-mathematics that would allow no distinctions in infinities beyond countable and uncountable? I know intuitionism is more restrictive about infinities than traditional mathematics, but it's way *too* restrictive for my tastes, I wouldn't want to throw out the law of the excluded middle.
I don't know. In general people who accept the uncountable accept most big cardinals. This follows from the fact that in most set theories you can prove Cantor theorem which say that card(power of A) is always strictly bigger than card(A).
In some "toposes" you can certainly build ad hoc models of set theory with a modified power set axioms making possible to collapse the uncountable sets (but that's whisfull ad hoc mathematics).
Note that by "Skolem Paradox" the notion of cardinality is shown to be relative. (I have use this to refute a set-theoretic analog of Lucas' godelian refutation of mechanism). That relativity is a reason for me (like you it seems) to be unplatonic with the notion of sets (unlike the notion numbers).
One way out is to have some constructive axiomatic where you modelize "uncountable" by "non recursively countable" ...
But for machine, you can guess, the big whole is not even nameable, and we (would-be computationalist) must learn to accept the plausibility of it. Set theories, by their failure, render that more palpable.