But proving the object exists is still useful, of course: it effectively means you can assume an oracle that constructs this object without hitting any contradiction. Talking about oracles is useful in turn since it's a very general way of talking about side-conditions that might make something easier to construct.

Of course. Though it's also important to note: whether or not an object exists is dependent on the logic being utilized itself, which is to say nothing of how even if the object holds some structural equivalent in the given logic of attention, it might not have all provable structure shared between the two, and that's before we get into how the chosen axioms on top of the logical system also mutate all of this.

It's not that classical logic is useless, it's just that it's not particularly appropriate to choose as the basis for a system built on algorithms. This goes both ways. Set theory was taken as the foundation of arithmetic, et al. because type theory was simply too unwieldy for human beings scrawling algebras on blackboards.