This bike is going to require a fair amount of patience due to the fact that it sat for so many years. My
CB360G sat outside and unused for 34 years (a decade less than yours has been sitting) and I took it down to the frame and rebuilt it for a variety of reasons, not the least of which was the need to transport it 600 miles in a passenger car.
Wasps and other bugs like these protected, dry environments, so you may find more of that as you clean out other areas. Mine had a mouse skeleton in the air box cover, but the Owner's Manual was still intact under the seat!
If you completely separate the housing from the bar, you might be able to get the he throttle cables out of the tube, but the wiring doesn't give you much wiggle room and it's a bear to pull those old hardened plastic sheaths through the bar. I think I brought my bar inside the house and spent a couple of hours doing that back in 2014. It wouldn't hurt to remove the throttle cables and lube them, off the bike. Or replace them.
There was some confusion earlier regarding the main ignition switch (key) and the ignition kill switch (RH control).
- You hear the click when you jump the solenoid leads to the battery terminals, but the starter doesn't get power. As mentioned above, this tells us the solenoid is not working. Either the contacts are corroded or the contact plate is not moving far enough to close the high current switch.
- The fact that you connected the leads of the solenoid to the battery suggests that the starter button was not working. You can test its function with a multimeter. Pushing the starter button should create a path from Y/R to ground. The starter solenoid receives power on the low current side from black and connecting Y/R to ground closes the circuit and throws the contact switch for the starter (in theory).
- The main switch delivers power to the harness when the key is in the On position. The black wire on the main switch receives battery power from the red wire in this situation. With the key on, you should see 12V or so on the black wire. This is necessary for the solenoid to function. Main switches do go bad, but I think it's unlikely here.
- The black wire ends up going through the ignition kill switch and provides power to Bk/W for the coils in the Run position. This does not affect the starter solenoid function.
- PJ tells us that your model is early enough that there is no clutch switch, so the starter can be engaged with the bike in gear. Avoid that.
Have you verified that you are getting power with the key on? Do any lights come on? Does the horn work? Etc.
Keep at it. The bike will start cooperating at some point.