Comment by Jweb_Guru

The reason people had these assumptions is because people have been trying (unsuccessfully) to find a constructive interpretation of this stuff for a very long time. Even very fundamental results in measure theory like the Heine-Borel theorem typically require some extension to traditional constructive axioms. Like I absolutely get where you are coming from, but there are a large number of "nice" classical results that definitely do not have constructive counterparts. It's cool that descriptive set theory is not one of them but it's not obvious by any stretch of the imagination, and the pattern you're using to say that it's probably true ("Curry Howard maps nice to nice") is not great process IMO since it would fail in a lot of other cases.