The question I had in mind was whether a mathematician would pause to say, of an integer that is completely unspecified, that he did not know whether it is prime or composite. What's usually wrong is that it is misstated. The theorems have computational interpretations, it's just that the computational interpretation is not justified by a nonconstructive proof. The latter has a much nicer computational interpretation than the former. I dare say that is why you phrase it that way.
Nobody likes clumsy theorems like the former.
I don't see why one would need that assumption. Of course if you identify truth with provability, as Brouwer appeared to do, then I guess that would be a consequence. I don't have to have a position on the nature of mathematical reality. Nevertheless, if you operate in a constructive framework, your view of mathematical reality will no doubt be colored by that experience.