> the vast vast majority of statements in any axiomatic system are going to be decidable

This is just flatly untrue, in the strictest possible sense, and even in the generous definition of "statements that mathematicians care about", that is going to be very heavily biased towards questions that are decidable, because the questions that are likely to be decidable are the ones that they can even reason about to begin with.

If you're a computer programmer, the CS shadow of Godel comes up _all the time_ in practice, and undecidable situations are quite common and don't really need to be carefully constructed.

This is just the drunk-in-the-streetlight situation. Things seem to be decidable everywhere we look because we look where things are likely to be decidable.