Are you sure the question was intended in such a philosophical manner? It can be deconstructed into ordinary high-school level physics:
Let's replace the car in our mind with any dumb block. The block "rests" (isn't moving in relation to our reference frame) either because there is no force applied to it at all, or because the force applied to it is less than the friction force. The applied force is not completely lost, it goes into internal deformations, even if we can't see it.
The applied force and the friction force are in static equilibrium. There is no notion of time, the force is either smaller than the friction, or it is larger.
Let's assume the force is continously increasing over time. Then there will be a momenent in time when the applied force overcomes the friction and the object would then start to be accelerated. Any infinitesimal amount of time after that leads to an increase in velocity v=a*t where the acceleration a comes from the applied Force a = (F_applied - F_roll)/m which is still reduced by the roll friction.
It felt to me that this was missing from the other answers. So again, time enters only because the force is changing. If it is not changing, the object/car either moves or doesn't move.
In the case of the car, it is the motor that is increasing the force applied to the wheels. There is no first moment of movement, any infinitesimal amount of time after the applied force has overcome the friction force results in a correspondingly small increase in velocity.
What happens between these 2 moments?
High-school level answer: The applied force has overcome the friction force.