F1 may or may not be equal to F2, but they do not constitute a Newton's third law force pair.
If you assume that the vehicle is not accelerating, then Newton's second law will tell you that F1 is equal and opposite to F2 (if you ignore any frictional effects). And if you assume that F1 is equal and opposite to F2 then Newton's second law will tell you that the vehicle will not accelerate. But you cannot prove that the vehicle will not accelerate this way.
Newton's second law has no application in a non accelerating reference frame.
Videos I showed you multiple times show that F2 = F1 and there is no acceleration.
In order for vehicle to move relative to ground one of the wheels needs to slip.
1.
No wheel slip F2 = F1 according to Newton's 3'rd law.
2.
Left wheel (output wheel) slip during acceleration period F1 = F2 + (m * a) thus both 3'rd and 2'nd law are added and F1 is larger so vehicle moves to the left (it is dragged to the left as there is no wheel rotation).
After acceleration stops and vehicle is dragged at constant speed F2 = F1
3.
Right wheel slip F2 = F1 while energy storage is charged so right wheel rotates but vehicle is not accelerated then stored energy is converted to vehicle kinetic energy F2 = F1 + (m * a) again both 3'rd and 2'nd law applies this time the acceleration is due to stored energy and so vehicle moves to the right.
All 3 cases where demonstrated in my videos showing the real world experiment.
Notice that in none of the cases the F2 = 2 * F1 that will be the case for a 2:1 gearbox. But there is no working gearbox as that requires 3 points of contact and will not work with just 2.