Some of those problems are "Nine women can't make a baby in one month" type problems. Sometimes the solutions are stumbled upon by accident, rather than by concerted effort, especially in medicine.
In fact it wouldn't surprise me if some sophisticated neural net algorithm developed for HFT was repurposed and used successfully for other things - medical research, climate modelling, brain research - basically anything involving non-linear dynamic systems.
Only problem of course is that all those HFT algorithms aren't FOSS, but the incentive to develop them in the first place would be greatly diminished since they'd lose whatever edge they had almost immediately after publishing.
The markets are a pretty decent lab for experimentation and learning about complex systems, which we're still not so great with, as long as systemic risk can be prevented and every player forced to take their losses instead of passing them off to governments. I personally don't think HFT is remotely as much a problem as what caused the financial crisis - a few flash crashes here and there plus an "HFT tax" on asset prices, vs the arguably-fraudulent origination and dissemination of trillions of dollars of bad debt resulting in the near total annihilation of the global economy. I think we'll gain a lot more (or at least lose a lot less) as a society by prioritizing the latter problem.