Cantor's diagonalisation is certainly constructive. Given a a sequence of sequences of binary digits, it gives a way of constructing a sequence not occurring in the sequence of sequences. The resulting sequence is as computable as the input.

But to answer your question: Many mathematicians regard their (classical) proofs as algorithms. They just allow themselves to use an oracle call for any well-formed question:

Say I want to prove B. Then, at some point in my proof, I formulate a sentence A, and make a call to the oracle. My proof/algorithm then branches: in one branch I assume A is true, in the other I assume A is false. If I can continue such each branch gives B true, then my proof works (classically). [0]

The constructive critique is that such an algorithm cannot be executed my humans, or turing machines for that matter.

Also there is a whole continuum of stronger and stronger oracles below this ultimate oracle allowed by classical logic — which is an interesting continuum to study. Often one can locate an exact class of oracles needed for a given classical proof to work out.

[0]: This is the «church encoding» version of the law of excluded middle: For all B, we have (A → B) → (¬A → B) → B, which is equivalent to the usual formulation (A ∨ ¬A).