First off it's not unsolvable, "particulates" aren't necessarily dangerous.
Yes in fact they are, or at least, any persistent particulates are dangerous. That's what makes automobiles so bad, and why DPFs actually make diesels worse. We've discussed here on Slashdot before that gassers actually make just as much soot as diesels, it's just far finer so it's much harder to detect, which is why this fact went unknown for decades. The reason it's hard to detect is that the particulate sizes are very small. When they get very small (PM2.5 and below in particular) cilia have a hard time removing them from the lungs and they tend to persist. The soot particles are very stable since they are made out of carbon. All persistent irritants are potential carcinogens.
What's certain is that tire particles aren't a guarantor of cancer .. it could at best bias the probability.. but not by much.
Some of the additives in tires are very carcinogenic.
it's likely not infeasible to make non-toxic tires.
It's both infeasible and impractical. Even the carbon black and silicates in tires can cause cancer for the reasons explained in the first paragraph. At best you can mitigate risk, you'll never make them non-carcinogenic. It would be better to also reduce the number of vehicles and also make tire compounds harder to reduce wear. This does reduce safe effective speed around corners and such, but most vehicles have a lot of excess in that department these days, and the ones that don't usually aren't going very fast. I like hard cornering, it's where the fun is in my opinion, but I do consider it to be more important to improve health.