Did you replace the PTO Switch?
If you did, did you replace it with the proper switch? The factory switch is a SPDT Switch. It is "on" in both positions. When it is in the PTO On position that set of contacts powers the PTO, and the Starter should not function at all. When it is in the PTO Off position, the switch completes the starting circuit to allow starter operation. Is the click you hear when you switch to PTO on the PTO operating? The electric PTO on the quietlines is very similar to the clutch on an automotive A/C compressor, so should hear a similar click to the A/C cycling in a vehicle. The "PTO Off" circuit is the orange one in the above drawing. The contacts in the PTO switch have to be closed in the off position, and you have to have your foot on the brake (or that could have been bypassed in the last 40 years) for the trigger post on the starter solenoid to get power.
If you didn't replace the PTO switch, try wiggling it when you are trying to start it. Every once in a while I have to do this on my Quietlines.... Hit the key nothing, wiggle the PTO switch the starter start spinning.
What condition is the igntion switch in? They like to rust and the prongs will get loose, or even break off. I had one with what looked like a good switch, but had all kinds of strange problems. Tried a different switch and everything cleared up.