The danger I see is mathematicians endorsing mathematics research because it serves explicitly mathematical goals....I'd like us to decide to attack [the Rieman Hypothesis] because we expect it to be useful, not merely because it's difficult and therefore allows us to demonstrate skill.

Why such prejudice against "explicitly mathematical goals"? Why on Earth is this a danger? One way or another, people are going to amuse themselves -- via art, sports, sex, or drugs -- so it might as well be via mathematics, which even the most cynically "hard-headed" will concede is sometimes "useful".

But more fundamentally, the heuristic you're using here ("if I don't see how it's useful, it probably isn't") is wrong. You underestimate the correlation between what mathematicians find interesting and what is useful. Mathematicians are not interested in the Riemann Hypothesis because it may be useful, but the fact that they're interested is significant evidence that it will be.

What mathematics is, as a discipline, is the search for conceptual insights on the most abstract level possible. Its usefulness does not lie in specific ad-hoc "applications" of particular mathematical facts, but rather in the fact that the pursuit of mathematical research over a span of decades to centuries results in humans' possessing a more powerful conceptual vocabulary in terms of which to do science, engineering, philosophy, and everything else.

Mathematicians are the kind of people who would have invented negative numbers on their own because they're a "natural idea", without "needing" them for any "application", back in the day when other people (perhaps their childhood peers) would have seen the idea as nothing but intellectual masturbation. They are people, in other words, whose intuitions about what is "natural" and "interesting" are highly correlated with what later turns out to be useful, even when other people don't believe it and even when they themselves can't predict how.

I believe one way to improve the peer review system would be to explicitly claim that your work is motivated by some real-world problem and applicable to some real-world solution, and back those claims up with a citation trail for would-be groundedness-auditors to follow.

This is what we see in grant proposals -- and far from changing the status quo, all it does is get the status quo funded by the government.

It's easier to concoct "real-world applications" of almost anything you please than it is to explain the real reason mathematics is useful to the kind of people who ask about "real-world applications".