I would say it's more accurate to say the controller regulates the ENERGY into the motor in terms of amps x time.
Using an extreme example, if you considered two rectangular pulse waveforms. Waveform A has a very high peak current and a very narrow mark. Waveform B has a very low peak current and a very wide mark. The area of both (and hence energy delivered) is the same but which one gives you the most torque?
For the posh controllers to maintain everything perfectly as the battery volts drop they would probably need some kind of DC-DC converter in there which is fairly unlikely given the size / efficiency constraints.
The controller is exactly that it is a class of a DC to DC convertor.. it pumps pulses of charge through each of the coils It could be classed as either a single polarity voltage input to 3 Voltage output regulated voltage outputs bipolar. Or a DC to 3 phase AC convertor or anything else you wish.. . Each charge pulse provides impulses of mechanical force.
.. The inductance of the coil in each winding, limits the rise in current as a function of time. It might be better to think of it as quantity of charge ( coulomb's ) deposited during each pulse. The current in any winding never reaches the steady state value that ohm's law would predict. It gets switched off long before.
A larger driving voltage will assist in pumping more charge down the coil, for a fixed fixed on period. The designer of the controller can vary both the pulse rate , the PWM pattern, and the duty cycle. Even with a scope, and viewing the voltage waveforms to discern the actual patterns would be extremely difficult. , As it needs the motor windings to integrate the pulses into a current waveform.