I wonder how analysis made it into mathematics. Now there is just a schism in the doctrine in a lot of places.
I think it is a "does it 'solve' problems I care about?” kind of thing. I think it is the foundation for validating inferences in math. The choice axiom went from, being considered problematic, to being mostly accepted because new problems from it did not hurt old problems.
Can be done rigorously now, wasn't done so for decades. It was tolerated, because such problematic symbolism was used anyway in physics.
I think anyway, such an area had to be pushed for a long time to make it into convention. Philosophy, not being a moneymaker for math, has had less luck.
Correct results? Ah you meant empirically. That in itself is has been argued as a basis by Putnam.
https://link.springer.com/chapter/10.1007/978-94-010-3381-7_5
It does not mean that there does not exist a constructive proof of alignment of JWST mirrors. Although a non-constructive proof of that last statement might be possible to give.
So no, not refusing to make them work. Saying we do not actually know how they work, or that they really do.
Empirical scientists and statisticians are satisfied with the mirrors. Some mathematicians are, others are calling everyone else too gullible. Solipsists refuse to entertain the existence of mirrors in the first place.
There isn't quite a consensus on the ontology. But the point is, that yeah, definitely seems that we are structuring our reasoning to get the answers that benefit us.
Putting our problems at the center does work though. Its a form of egoism and has been around almost as long as boolean algebra. 
https://theanarchistlibrary.org/library/max-stirner-the-ego-and-his-own
Anyway I don't want to refer to the results of computation as empirical since that's begging the question in favor of realism, though I think the realist arguments are convincing.
The crux of this topic is whether the human mind can reach out and describe all of reality with mathematical/scientific models, and the result we consistently find is that we can't describe everything all the time everytime, but we can very well describe some subset of things most of the time and that those constructed descriptions tend to line up with our unconstructed intuitions.
Also tangentially I find every discussion about the parallel posulate to be annoying when they draw two Euclidean lines then try to tell me that it's something else. If they drew instead drewtwo projective lines converging to a vanishing point then it's obvious that all parallel lines intersect at infinity, because that's whats been drawn.