Difference between capacitor start and capacitor run Induction motors?
Why not experiment with a capacitor-start motor yourself to see what happens if the start capacitor remains connected after the motor has reached operating speed.

You would notice two or three things:

1) the motor starts to get very hot.

2) the motor does NOT reach full speed until the starting capacitor / winding is disconnected.

3) the motor makes unpleasant sounds.  Sort of a growling sound.

Like you, I was curious when I was a youngster.  So I experimented to find out what happens.

That experience has done me well over the years - it makes diagnosing a malfunctioning motor easy.