See Andrej Bauer's proof that they are indeed uncountable: https://mathoverflow.net/questions/30643/are-real-numbers-co...

I'm still unconvinced.

Is there a specific issue you have with this proof? With respect, I think you may be conflating a constructive proof of the uncountability of the reals with a proof that the set of all computable reals is countable. These are two different claims.

A statement about the reals which is provable under constructive mathematics does not reduce to a statement about the computable reals and vice versa. A set can be constructively definable without all of its elements being constructively enumerable.

Given any explicit sequence of real numbers, it's always possible to generate a number not in that sequence. In fact, this can be done algorithmically using essentially Cantor's method.

A simple diagonalization argument is perfectly constructive.

Yes, and to clarify what I predict might be a point of contention about the necessity of "contradiction" in Cantor's argument, some of the comments in this thread might be helpful: https://reddit.com/r/math/comments/70xeon/why_is_cantors_dia...