88mph is an important speed to screw the space-time continuum. But, motion is relative...
Hey doc, can you please explain 88mph relative to what?
What we see in the movie: it was the speedometer reading which matters. So, place the car on a conveyor belt moving in opposite direction of the car having the car's speed and acceleration. The car will reach 88mph without resultant displacement and will go time warp.
To confirm, the speedometer was normal without any modification to record absolute speed (which is impossible). You can see in the first time travel (of Einstein) that the remote-controlled car was slipping on road initially but still Doc was recording increasing speed.
This means it's clear that the time circuit needed only a revolving thing in space with a tangential speed of 88mph (w.r.t. the axel which is, at rest, relative to the time circuit). In the case of the flying car from 2015, a similar setup in rocket exhaust can be assumed (motion is relative, after all).
My question: why didn't Doc use a time machine at rest with revolving parts? It would be less risky. (Note: Marty crashed twice.) Plus... instead of pushing the car with a steam train, the revolving parts could be revolved easily with a steam engine.