I have a similar problem with my 17 HP craftsman. Battery reads 12.3 volts with key off. Brand new battery. New solenoid. Replaced starter with used part. When new battery is put on mower cranks right up. However after the mower has been started a few times it drags the battery down and starter makes a whining noise and won't crank. While mower is running the battery charges slowly. Went from 12.3 volts off to 12.7 in about 30 minutes. Voltage at starter, stator, and solenoid are good. Once started mower runs fine, just doesn't seem o have enough juice to turn the engine over after a couple cranks. I'm leaning towards ignition switch? Any ideas?

Hi,
Moved your post for visibility. Could easily be that the contacts in the switch are corroded. Could also be the health of the used starter. What does a voltmeter read at battery teminals with engine running at full speed?
Peace,
Clarke