1. Accepting versus rejecting the axiom of choice.

2. Classical versus intuitionistic mathematics. Intuitionism goes much further than just rejecting the Axiom of Choice, and e.g. says that you haven't proved "p or q" until you have either proved p or proved q. It denies not just the axiom of choice but the "law of the excluded middle" which says that for any proposition p, either p is true or not p is true.

3. Accepting versus rejecting "large" sets -- in your case, you say we should be suspicious of anything that assumes the existence of uncountable things. (I think this is a bit unusual; there are finitists who deny that there are any infinite sets, and ultrafinitists who go further and say there aren't even arbitrarily big finite sets, but it's not so common to accept countable but not uncountable infinities.)

I think intuitionism implies rejecting AC. I don't think any of the other possible implications between these three things holds; e.g., so far as I know most intuitionists have no particular problem with the existence of large infinite sets.

(Actually, probably some theorems of the form "if there are no large sets, then the axiom of choice is true for boring reasons" are provable.)