A question about using PID to improve transient response for a DC motor
You are correct, even a PID loop can only apply 100% duty cycle, so maximum speed transient response is not improved.

The difference is if you wanted to go at 0.75 of maximum speed, then the PID loop will apply more power at the beginning than the 0.75 duty cycle and then lower down to the appropriate duty cycle as the error gets closer to 0.  In this way, the transient response is improved.

Edited to add:
Also, after reading your question again, you can only use a PID loop when you are controlling motor voltage using PWM or some other method of controlling the voltage/current to regulate speed/current/power.