I've got a 23 hp Kohler also. Starter was messing up and finally got to where it wouldnt engage at all. I had to take a screw driver and stick it under the cover and raise up the drive gear on the starter while i hit the switch. Not a good way of doing it but it was the only way to start it untill I bought a new starter.
And on a side note I ordered the starter online because it was about half the price compared to a dealer. Now it occasionally hangs up and have to take a screwdriver and get the gear down,
Should have just paid the extra and got it here locally.