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Most of the Zero Turn Radius (ZTR) machines have a circuit breaker in the system. Sometimes that breaker is tripping and that's the "tick" you hear and it's easy to confuse that with the "tick" a sticking solenoid can make. I've seen the circuit breakers fail before causing a similar issue. It's worth mentioning that the "tick" you hear could also be the fuel shutoff solenoid on the carburetor.

Outside of all that, in general terms if the machine can be started by crossing the solenoid poles that's usually caused by a failed safety switch. That could be a safety switch itself, the wiring going to it, the PTO switch, or even the ignition switch. All of those can cause that symptom. From my own personal experience I have seen the fail frequency in this order:

1. Safety switch
2. PTO switch
3. Ignition Switch
4. Harness

You've replaced the solenoid and harness so I have no reason to suspect that either of them are the problem at this point. I would get a meter and meter out the safety switches to make sure they're working as designed. You just need to check for continuity across each of them. If they all check out good, I would then have a look at the PTO switch, then on to the ignition switch. On a realistic level, if the safety switches check out good, it might not be a bad idea to have a technician come out to have a look at it. Sometimes chasing an electrical gremlin around on a machine can be a frustrating way to spend an afternoon.