I can understand some of that discomfort. However, ultimately, uncountable infinities and (the axiom of) choice leads to other interesting scenarios that, IMO,

I can understand some of that discomfort. However, ultimately, uncountable infinities and (the axiom of) choice leads to other interesting scenarios that, IMO, it's best to make peace with. Besides, mathematicians rarely study arbitrary, probabilisticly-common, objects. We focus on objects with “nice” algebraic properties. Similarly, it doesn't matter that almost all numbers are uncomputable when we don't use them in practice.