This is really a quibble turning on how one defines “constructive”. Let me propose two scenarios, which I will call NC and C. I believe that both scenarios are epistemically possible (given current generally available knowledge), but we do not know if either is logically possible.

In scenario NC, a mathematician devises a highly novel, unexpected and correct proof that P=NP, by finding a way to translate that statement into a statement in some seemingly unrelated area of mathematics, and then using some very deep and profound and innovative techniques to prove that statement in that seemingly unrelated mathematical area. Now, suppose those techniques are fundamentally non-constructive, in that they rely on methods of a kind of which no consistent mathematical constructivist could approve. This kind of proof would not directly turn on finding some novel algorithm for some NP-complete problem and directly proving that the algorithm executes in polynomial time. The only algorithms invoked may be some relatively straightforward algorithms needed to translate P=NP into that very different mathematical area, and then the real genius of the proof may be done in that other area and not explicitly rely on any algorithms at all. This would be a fundamentally non-constructive proof.

By contrast, in scenario C, we have a very different proof that P=NP, which might have as its centrepiece some very novel and obscure and maybe even bizarre algorithm, along with a proof that the algorithm correctly and exactly solves some NP-complete problem, and also a proof that the algorithm executes in polynomial time, and those two proofs may depend on quite subtle details of that specific algorithm. Let us suppose that the profound mathematical genius who devised this algorithm and the associated proofs has great sympathy for mathematical constructivism, and hence does not use in the proof the kinds of techniques of which a mathematical constructivist would disapprove (or, at least, not directly). Maybe even, those proofs are relatively straightforward once you have the algorithm, and coming up with the algorithm was the real genius of the overall proof

Now, you’ve pointed to an example of an already well-known algorithm which solves an NP-complete problem in polynomial time, but only if P=NP. And you are claiming that turns any non-constructive proof into a constructive one. But in scenario NC, our proof was non-constructive in two ways (a) unlike the proof in scenario C, this proof does not have as its centrepiece the construction of a specific example of a novel polynomial time algorithm for solving an NP-complete problem; (b) and unlike the proof in C, it relies on mathematical techniques of which a mathematical constructivist would disapprove. When we combine it with your proposal, it does not alter either of those two ways in which the proof in scenario NC is non-constructive, and so does not actually convert a non-constructive proof into a constructive one as you claim it does.

The term “constructive” has multiple meanings, and even if your argument succeeds in converting a “non-constructive” proof into a “constructive” proof in one sense of “constructive”, it fails to do so in other important senses.