I'm unsure what you did to bypass. Maybe you could enlighten us? Use specifics. The PTO switch itself has a safety side running through it and possibly was damaged at the same time as the PTO cooked itself. No stray ground wires left off during the repair?

I have used this bypass procedure on other pieces of equipment. Run a wire from the positive side of the battery, install a 15 amp fused link, available from Oreilly auto parts, napa, to one side of the switch you install in the dash. Run a wire from the other side of the switch to the pto. I cut wire that used to go to the pto switch and soldered a wire to it which goes to the other side of the switch. Necessary to get away from the regular PTO switch that came with the machine.
Attach an eye terminal to a 14 or 16 gauge wire, attach a fused link, I used a standard fused link and installed a 15 amp fuse. Attach the other end of the wire to one side of the switch. Cut the wire that powers the pto and solder, attach a wire to it and run it to the other side of the switch.

"Does the tractor have to be running prior to the pto switch working?"

Not if the PTO is wired correctly. In your case with the switch you installed.

Hot (+) wire from battery to switch.
Wire from switch to the Hot (+) on the PTO.

The PTO will not turn unless the engine is running. But when you pull the switch. You should hear it and be able to see it engage. Same goes for the PTO if all of the safety switches are working and you engage the blades with the key on.

Received a used ignition switch with the control board from ebay. Installed the switch and the pto now works correctly. It will engage with switch on and brake in lock position. A distinct engagement noise is heard when the pto switch is engagIed This particular machine has required a lot of repairs. A very well build piece of equipment. LX188