Isn't the whole purpose of the circuit so that when joining on downwind (from overhead, deadside or direct) you've got the time to assess who is in the circuit and where, and not to interfere with traffic on final and upwind, so that on final and upwind you can concentrate on flying the aircraft? If you've got a burst tire or another reason to perform a go-around (or not), the least of your worries should be an aircraft on downwind. No matter whether he's got a radio or not.
And as for vacating the runway, same as Crepello says, blocking the runway for 15 minutes should not be the concern. Keeping the airplane in one piece is far more important. So while the aerodynamic controls still work, I'd use them to stay on the runway (in case of a right tire burst, even steer left of the centerline if I have the spare mental capacity) and as soon as you're below flying speed, stop as quickly as possible. On the runway.
Personally I think I'd do a quick pan call, shut everything down and evacuate. Sure, in a 172, solo, you're out in just a few seconds (I fly PA-28s and DA-40s which are a bit harder) but next time you might have passengers with you as well. And remember it only took one chunk of rubber to puncture the fuel tank on that Concorde. Better evacuate first and find out a minute later that it would not have been necessary, than waiting too long and becoming trapped. Plus, as you said, taxying was impossible even with full power (useful to know, BTW) so you have to get out anyway.
Just FYI, we had an accident with a DR-400 (Robin) a few months back. The DR-400 had four persons on board and a bit more of an aft CoG because of that, than the pilot was used to. He did a perfect landing but because of the aft CoG the nosewheel got into the steering lock. He slowly veered off the runway, couldn't steer, so he tried a go-around. Hit the lights (exit lights I think) at nearly flying speed, causing a lot of damage. But if he would have breaked, even with the nosewheel lock engaged, he would have stopped either on the runway or just off it, in the grass with virtually no damage (other than a bruised ego). And - but he didn't know this - would have freed up the steering lock and gotten directional control back.
So for me, if something strange happens on the runway and I have runway to spare (which, in a light aircraft, is almost everywhere), I stop. Sorry for the others that have to divert or hold, but it's my aircraft that's acting funny, not yours.