Firstly: this can probably be neither proven nor disproven, but my intuition tells me that it's by definition impossible for mathematics to run out of problems.

Secondly:

> It cannot remain healthy with its incredible publication rate today of mostly useless generalizations.

So the issue isn't that mathematics is running out of problems. The issue is that there are more publications than there are new problems being discovered / solved, and, ergo, the majority of publications are of limited value / interest. And that isn't an issue unique to mathematics, that's just how academic research is in the 21st century!