Doesn't matter if you use it, it works anyway...
It doesn't "work" if you can't calculate it, as in the case of the gravitational slingshot our interplanetary probes regularly use.
I don't understand what you mean. Are you saying the car looses grip as the wheels spins faster, or are you just saying it takes energy hence power to spin up the wheels? As far as I know neither would explain almost halving the acceleration, as in your example graph.
The rotation rate of the wheels is proportional to velocity. The force of acceleration delivered is inversely proportional to wheel rotation rate. If the velocity is doubled, the force of acceleration applied to the vehicle is halved for the same input power. That's why acceleration drops off.
That doesn't control how much power is needed to achieve a greater acceleration. The wheel rolls at the same rate at any given speed. Getting to that speed faster takes more power. What is the ratio of power needed at a given acceleration?
P = a·v = a·(a·t). Higher a means smaller t to achieve a given dv, which is linear. But that's only the v=a·t term, while power is a·v. So for an arbitrarily small dv or dt, the instantaneous power required at any moment is proportional to a². The math works.
When both P and v are known, then a is limited by P. If a and v are known, P is derived and possibly unlimited (for example, free fall in a gravity field).
Physics does not stop working just because you don't believe in it. It obviously takes more power to accelerate at higher speed, or with constant power you accelerate less at higher speed. Just as your graph shows.
That's why I trust the math showing how the physics works. If I have to supply all the energy, then only conservation of energy applies. For a rocket, it is conservation of momentum that dominates. Reaction mass goes one way, momentum makes rocket go the opposite way proportional to mass.
The energy needed is calculated by conservation of energy in the acceleration of the exhaust mass. We use an exothermic chemical to do that rather than a mechanical acceleration. Because the exhaust velocity is greater than the dv imparted to the ship, much of the energy is wasted compared to some hypothetical means of using energy to directly accelerate the ship.
Consider the heat engine. Using electricity to directly heat up a resistor is the least efficient means. Using electricity to drive a pump to move a working fluid taking heat from one region to another is more efficient. Run a compressor to take advantage of state change is more efficient still, able to take heat from a cooler region via decompression to a warmer region via condensation.
A grav slingshot uses essentially no input energy and takes energy from the gravitational body that changes the probe's course. Grav drive would do something similar, pushing against the fabric of space and taking energy from any nearby bodies.
