Recall that the set R of real numbers R is defined as the set of numbers that can be shown to exist, plus an uncountable collection of "numbers" that are assumed to be near the set of numbers that can be shown to exist.
It's not amazing that assuming something unnatural (Uncountable Choice) gives an equally unnatural consequence. The set R of "Real" numbers is a fantastical "object" that has fantastical properties. It can do apparently impossible things because ZFC flat out assumes that it can do apparently impossible things.
IIRC the real numbers are, roughly speaking, more-or-less isomorphic to infinite sequences of digits. If you believe real numbers are fantastical, I suppose you also believe infinite sequences are fantastical? We can't have one in the universe, because we'd run out of atoms, but we can't have circles either and that is rarely complained about.
Historically the definition of computability came quite a bit later than ZFC. Zermelo might have formulated his set theory differently if computability came first.
It's not amazing that assuming something unnatural (Uncountable Choice) gives an equally unnatural consequence. The set R of "Real" numbers is a fantastical "object" that has fantastical properties. It can do apparently impossible things because ZFC flat out assumes that it can do apparently impossible things.