Bad solenoid? If you jumper across the solenoid terminals will the engine turn over? Load tested the battery or tried jumping from a good one? Lights coming on doesn't necessarily mean "good." Can you turn the engine over by hand? e.g. hand on the screen on top of the flywheel. Any noise at all when you attempt to start it, "whirs, clicks, etc?" Check the oil for presence of gas (hydro lock). Starter circuit is very simple, +12Volts to the coil of the soenoid, contacts close and +12V from the battery to the starter, admittedly, the "permissives" (all the switches you've replaced) come into play. Also seen key switches or connections go bad. Either preventing the solenoid from being energized or if the engine is running to interrupt the ignition killing the engine.

I'd try jumping across the solenoid contacts first, or just take a jumper wire (at least 10AWG) and put +12V onto the starter terminal and see if the engine spins. If successful, move back from there and jumper +12V to the coil of the solenoid, if it clicks and the engine spins, something is not permitting (permissives) the voltage to get there. Also check to see if the mounting bolts of the starter are tight as they are the ground connection. If you don't have a meter, now would be the time. Actually, a test light which indicates presence or absence of +12V would be adequate.

Good luck,

Ev