Car wheels have an entirely additional job besides supporting the chassis that aircraft don't: transfer engine power into motion.
Aircraft have only one: converting rotation into heat. Other than during deceleration the wheels spin freely. Whereas on a car larger wheels mean lower speeds in the transmission, gearbox and ultimately crankshaft. (You can try this by changing the size of the wheels on your vehicle and observing how the speedometer now needs to be recalibrated)
Brakes themselves are pretty good and don't dictate the size of wheels once you cross a certain threshold.
Another key factor in wheel size is ground clearance. The axle is often the lowest clearance on many vehicles (suspension lifts are popular in off-roading world which is slightly bizarre given it does nothing to change the minimum ground clearance unless the sump or gearbox already hangs below the axles). This isn't really a concern in aviation, in part due to the fact there aren't axles connecting e.g. left and right main gear, the wheels in each set function almost like a single wheel in terms of ground clearance. (and the clearance is really there for the engines rather than particularly rough terrain)
So it's not just a case of compromising on requirements between use cases - some of those requirements don't even exist in both cases.