I doubt that scientists posit that singularities "aren't natural objects"...not even sure what that term means, if anything.

I think you have it right, the term singularity is a placeholder for an event/phenomenon that we think happened but can not observe and have no hope of ever observing,

That doesn't mean that we can't imagine and propose a scientific explanation (one based on prior observation), just that we almost certainly can't test it.

A lot of quantum physics is similar, suggested by observations and maths, seeming to check out in subsequent observations, and incredibly difficult to actually observe and thus test.