> My point isn't that most algorithm/program proofs need the excluded middle, it's that they don't benefit from not having it, either.

Because if they benefited from it (in surfacing computational content, which is the whole point of constructive proof) they'd be comprised within the algorithm, not the proof.

> in surfacing computational content, which is the whole point of constructive proof

The point of a constructive proof is that the proof itself is in some way computational [1], not that the algorithm is. When I wrote formal proofs, I used either TLA+ or Isabelle/HOL, neither of which are constructive. It's easy to describe the notion of "constructive computation" in a non-constructive logic without any additional effort (that's because non-constructive logics are a superset of constructive logics; i.e. they strictly admit more theorems).

[1]: Disregarding computational complexity