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 p

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.