I would try a new battery first. When they get weak, the starter won’t fully engage. if a fresh battery doesn’t solve the problem, then look at the gear and make sure it moves smoothly.

Batteries can seem good and not be. If you have a battery store near you, find out if they can load test your battery. Another tet would be to charge the battery, remove the charger and check the voltage. Wait a day or two and check the voltage again. Should be at least 12.5v.
Check the teeth on the ring gear. Pull the bendix off and check that it’s not gummy or binding up in any way.

Battery buying note. I’d stay away from buying a battery from a battery store or any low cost battery in general. They won’t stand up to use on a track. Have a couple that didn’t last a day. Go to Comet or Russell or other large kart part distributer. The more expensive battery will cost you less in the long run.

Sounds like the battery first, then if that doesn’t cure it, the bendixes are notoriously bad.  I’ve replaced many of them.

Seb,
You still might need a bendix as Tim mentioned. You could try to determin if it fails at one particular point on the flywheel, but doubt if you’ll find anything.
I’ve sent you a PM.