Ok, when I say mathematically consistent I don't mean it in an axiomatic sense. In fact, like you point out there is a lot of unanswered questions even within quantum field theory whether or not it's mathematically well defined. In fact, one of the millennium prizes is related to this.

When I say mathematically consistent, I mean it in a looser sense. If we take a step back to before string theory, there was no way to get consistent results from quantum gravitational calculations. The usual tools that we use to renormalize quantum field theories do not work for gravity. This suggests that there's some type of "ultraviolet completion" of general relativity. In other words, the theory of GR ought to come with some implicit energy cutoff beyond which the theory somehow changes. String theory is such a change in that the stringy corrections to GR would only come into effect around the Planck scale. It's by no means necessarily the unique such completion, but as of now it's the only one we know of.

As to your other point, I think it's a good idea to reframe the work done in string theory as (what I used to joke) "theoretical theoretical physics". In other words, it may be the case that string theory is a true theory of nature, but even if it isn't, the theory lets us explore what such consistent theories could look like and how various paradoxes (like the information paradox in black holes) get resolved. These types of insights may point us in a direction of further investigation that may very well fall outside string theory.

In other words, at the very least (and I personally think this is underselling string theory by a lot) string theory is a proof of concept and a very powerful, sophisticated, and rich arena in which we can begin to understand the salient features of quantum gravity. One such example is AdS/CFT.