If your point is that it isn't necessarily useful to try to say in what sense our procedures "correspond," "represent," or "are about" what they serve to model, I completely agree. We don't need to explain why our model works, although some theory may help us to find other useful models.

But then I'm not sure see what is at stake when you talk about what makes a proof correct. Obviously we can have a valuable discussion about what kinds of demonstration we should find convincing. But ultimately the procedure that guides our behavior either gives satisfactory results or it doesn't; we were either right or wrong to be convinced by an argument.