Do any aircraft automatically test that the pitot-static system is functioning properly before liftoff?
First of all, ground speed is already determined by the IRU’s. No need for landing gear sensors although most airplanes do have WHEEL SPEED transducers for anti-skid currently installed. Wheel speed is not always equal to groundspeed but can be at times. For example, wheel speed right after takeoff is different than ground speed because the wheel begins slowing the second it leaves the ground.

Second, airspeed changes with altitude. So, the higher you are the faster your airspeed. This means that taking off in Death Valley might show a different airspeed RELATIVE to groundspeed than in La Paz El Alto. 

Third, most of the time, airspeed doesn’t register correctly until about 60 knots. You’ll see that most airspeed indicators have 60 knots as the lowest indication mark. This is because if the airplane is pointed into the wind, it could register spurious readings. So, before any checking gets done, the plane needs to be moving above 60 knots.... Like during takeoff.

Fourth, airspeeds are cross checked during takeoff roll. The vast majority of airspeed splits on a new flight leg occur on takeoff roll and result in a rejected takeoff - your premise anyway. Occasionally it’s not noticed until takeoff. This is rare. The other instances (which I work on more frequently) are in flight malfunctions usually related to inadequate pitot heating.

Fifth, the current system works very well. No need to reinvent the wheel.