speed of problem solving in grad school in mathematics
Success in well-designed qualifying exams in mathematics will depend very little on "problem-solving" speed or talent, but, rather, will depend on whether you've already done problems nearly identical to the problems which appear.

That is, as you observe, it would be prohibitively slow to "solve" many of those problems "in real time". Thus, that is not the expectation. Rather, the "test" is whether examinees have studied sufficiently broadly so as to have seen examples resembling the instances occurring in the given exam.

And, no, it's not about "memorization", either, which tends to be insufficiently flexible to allow easy adaptation to slightly changed situations.

So, really, it's not about "speeding up in problem-solving", but to be able to merely "remember" instead of "solving".