My definition of "running out of problems" as I stated was "running out of problems that more than a handful of people care about", and I think this is definitely true in math.
I honestly think Math as a field has always been defined by "problems that only a handful of people care about".
The only exception I can think about is maybe basic addition and multiplication.
Simply put, it's very hard to predict the usefulness of math at the time it's created.