I just finished our 1999 suzuki 300 king quad. Same exact symptoms.
Got to the starter and bench tested it with a Battery charger and it worked sometimes and not others. It wouldn't work at all off of the quad battery which was fully charged.
Took the starter apart and the I think it's called an armature (The copper thing the brushes touch) was covered in grime and residue from the brushes. One brush was stuck because of the grime in it. Sand papered the armature nice with extra fine, got it all cleaned up. Blew it out with Compressed air. Didn't even have to change the brushes. And it started right up.