First off, I am not advocating for rubber-bladder roads.
Car tyres mainly serve one purpose. That is, to provide a cushion of air for the car to ride on, thus reducing drag. They are however prone to flats as when driving behind a bumpy ox-cart laden with nails and screws.
My proposal is to do
away with the Achilles heel of the tyres, that is, the puncturability of the rubber shell. The tyre assembly would be created out of metal, the type of which would be determined though countless hours of research. Suffice it to say, it should be very resilient, non-brittle, common and cheap, fairly lightweight, and have a decent memory. Picture a nautilus shell with it's nested segments. In order to create a regular circle and not an ever increasing spiral, each segment would be slightly pitched so as to nest into it's neighbour all the way 'round. These segments would be slightly dynamic in that they could shift perhaps a millimeter back and forth. All segments would be joined to a standard wheel. (Imagine a leaf spring with a bit more curve to it). The overall shape would be that of a standard rubber tyre.
The tyres would not be subject to sharp pointy things or extremely sharp cattle guards. They would still gain the benefit of creating a cushion of air for the car to ride on through their slight elasticity. The design would be engineered to work similarly to bicycle spokes, that is, some would hold the hub in the air while others would keep it from falling, all through the strength of the assembly.
Now, where is the traction you might ask? Snow and ice might spell doom for your drive, resulting in dented tyres. Depending on which way the tyres were mounted, they could aid in either excellent stopping or excellent braking as the lip of each segment would provide traction. Perhaps your drive wheels would be lip down (as seen from standing in front of the car) and your brake wheels would be lip up. Perhaps more ideal for front-wheel drive cars. For added traction, each segment could have a rubber coating on its outer surface. Tyre chains may only need slight modifications to work with the metal tyres.
A master locking pin would hold the tyre together (similar to tyre chains) and dents could easily be changed by the layman. Jack up the car, remove the master locking pin, let the tyre clatter to the floor. Replace with new tyre, replace pin, lower car and drive off. "What if someone pulls my pin", you might ask? Well, don't blow up. Such pranks would be no more common place than a well aimed knife into the sidewall of your rubber tyre.