and swap the blade attachment if you are planning on using a new engine. I would go with a Kohler personally.
I would suspect a top bearing on the Onan causing the gear and starter to be damaged. Or the teeth are not a match. Although, 20 starts a day is a lot of wear and tear. I would check how the teeth mesh together by distance and adjust if able.
Go to the Honda site and see if they have a cross reference chart for the engines. Some places sell plug and play wire harness for the transfer.