The only reason F-1 is sticking to tiny 13 inch wheels and high sidewalls is the rules that mandate them. If more sensible sizes such as 15 inch wheels and same overall tire dimensions were permitted, the current wheels will soon go the way of the dodo.
In this era of CFD and ever more detailing to squeeze ever tinier gains out of an ever more over restricted car (courtesy of the Federation of Idiotic Arses), high sidewalls are a REALLY bad thing, especially when the sidewalls bulged under load in ways that completely screw up all those airflow interactions optimized through countless man hours and $$$ of testing. One example of this is the OMEGA tire from Michelin back in the good ole Michelin vs. Bridgestone days. The tire in isolation was worth almost a second per lap, but it was culled because its sidewall bulge characteristics interfered with the airflow so much the car ended up more than 2 seconds of the baseline (car with same aero package and conventional tires of the time) pace.
A tire of the same dimensions as currently used but with 15 inch wheels and more sensible sidewall height would obviously benefit from better tire dimensional stability, much better steering response and superior overall grip. In fact, Bridgestone engineers bluntly stated that a 15 inch version of current F-1 tires could easily end up almost undrivable due to G-LOC. This assume similar to current levels of downforce of course.
Remember the Renault mass damper debacle? The reason they used such as device was to cancel the very much unwanted harmonics from tires, which are underdamped.
In GT cars, 18 inches is both mandated by the rules and happens to be a very optimal size. Note that no one is silly enough to go back to smaller wheel, higher profile sizes these days, mostly due to very obvious technical reasons. Racing is the automotive equivalent of war, and in war, only victory counts.
