I said in classical physics, with is wrong, it propagates instantly.So the charge-motivating factor (I guess we're calling it "electric field") propagates instantaneously?

Maybe kind of like Bell's entangled photons then? Surely there is somebody out there who will argue with you about instantaneous propagation, but it strikes me as sensible.

In reality, it propagates at the speed of light. Do a calculation of how long that is for the capacitor you want to analyze and see if it makes a difference in any calculation you would do.

We use approximations all the time when doing calculations. The approximation of 0 for the time to propagate across the plates of a capacitor is a good one for any practical calculation.

Bob