It doesn't work like that. The effect of the downwash would be minimal in terms of added weight on the tires. But it would seriously disrupt the "clean" air that the car wings need to generate maximum down-force.
They need clean air in mid to high speed corners (where donw-force is of the essence to keep the speed). In fact one of the current problems that makes F1 boring (as in less overtakes) is that cars cannot follow each other closely for extended periods.
If a car follows another closely for 3 or 4 laps, the tires start suffering from overheating (due to the reduction in down-force caused by the disturbed air the tires have less downward pressure and grip making them slide more). After 3 to 4 laps following closely, the chasing car is forced to slow down to cool the tires and avoid damaging them.
