I think there are not many mathematicians who really believe that a non-constructive proof is worthless.

One reason for this is that, generally speaking, non-constructive proofs can in fact be embedded into constructive logical systems.

While that may be a reasonable justification for what mathematicians do, I think it is false as a historical claim about what caused mathematicians to do what they did. Mathematicians settled on their foundations ("No one can expel us from Cantor's Paradise," 1926) before they understood power and limits of constructive methods.

I'm curious if you are making a practical claim or a formal one.