Is it possible to store flyback voltage in capacitors?

Yes, it is possible to store that energy in a capacitor; after all, that's exactly what flyback-mode switching power converters do! So your idea isn't silly at that level.

However, using that energy to "boost" the motor drive is a little misguided, because the amount of energy you get from the flyback effect (which is due to "leakage" inductance in the motor) is orders of magnitude smaller than the amount of energy required to accelerate the rotor to any signficant degree. In other words, the effort required to implement this would not be rewarded with any practical benefit.

You may be thinking about regenerative braking systems. But such systems are not storing "flyback" energy — they're actually using the motor as a generator and storing the energy created by physically slowing down the motor (and the attached machine or vehicle, etc.).

In principle, yes. In practice, already doing it.

Consider the workings of a flyback diode. When the current to a motor is turned off, the energy in its windings (i x i x L / 2) needs someplace to go, and without the diode will produce a high voltage spike (V = L di/dt). The energy which needs to be transferred is shunted back into the motor power supply by the diode, and is soaked up by the output capacitance of the supply.

So the idea is not totally without merit - it's just that it's already happening.

It would be possible to capture inductive flyback energy in a capacitor, but doing so will make the motor run less smoothly and is thus only desirable if the intention is to induce vibration in the motor shaft. Instead, it's much better to allow flyback current to flow as freely as possible with minimal voltage drop, and use a fast enough PWM rate that the current doesn't rise or fall too much on any given cycle.

Basically, what happens is that during the "on" part of the cycle, the applied voltage works to get current flowing in the motor, imparting energy; the current can't stop flowing until the energy is dissipated somewhere. To instantly stop the current flow would require taking the energy out of the motor electrically. Allowing the current to keep flowing, however, will allow most of the energy to remain within the motor and continue doing useful mechanical work.

In general, the faster you can switch the motor current, the better. Unless you go so fast that transistors can't keep up, higher switching speeds will make the motor run more smoothly, efficiently, and predictably.