Maybe, maybe not. Let me put another twist on this. It will take a given amount of power for the starter to turn-over the engine. Weak batteries generally produce less voltage under a load such as running a starter. To produce the power required for the starter to turn-over the engine with a weakened battery producing less voltage there would have to be more amps running through the starter and the increased amperage may cook the starter worse than with a fully charged battery. Just a theory.

Similar situation when you jump a car with a dead battery and let the alternator on the car charge a dead battery. The alternator tries to put out its maximum amperage capacity for an extended period of time. It is an acid test for an alternator.

Be nice to your car and don't ask it to do these things, maybe it will be nice to you. That was the scientific part of the message.