This is of the latter kind. The usual set theoretic foundations is very powerful, coherent, concise, successful, explanatory, impressive, and totally dominating at this time. Taken as a whole, with the major supporting classical developments, it is certainly one of the few greatest acheivments of the human mind of all time. However, it also does not come close to doing everything one might demand of a foundation for mathematics. At the present time, there is no full blown proposal for scrapping it and replacing it with anything substantially different that isn't far more trouble than its worth.
Present cures are far far far worse than any perceived disease.
Now this does not mean that the usual set theoretic foundations might not give way to a better foundations, or might not be altered in some very significant and permanent way. In fact, I can tell you that I work on this from time to time. Sets under membership form the simplest foundationally complete system. Now before coming back to this, let me summarize the greatest of the usual set theoretic foundations of mathematics. First of all, set theory is unabashedly materialistic - a perhaps nonstandard word I use to describe the opposite of structuralistic.