Provided that wiring schematic is the correct one then the starter solenoid should engage when the deck switch closed, Pto Switch in off position, and the brake applied. The reverse switch has nothing to do the cranking other than killing the magneto while the engine running and you put the mower in reverse while the PTO is engaged,
Something is simply wired wrong and you need someone that knows how troubleshoot electrical problems. Plug and Play won't cut it. Admit you need some help from a knowledge person their hands on the mower. There is no shame asking someone to put their hands on the mower to help you out.
I have had many a DIYer come to my shop after spending a lot money trying to fix an electrical problem to only find something missed wired or fifty cent terminal bad.