I'm going to take a shot at this...
1st Question - Is the starter solenoid on the starter or fender?
In either case, if you don't hear the solenoid clicking, then it's not being energized....hence, work back from there. If the solenoid is on the right fender then it's easy to use a voltmeter to test for voltage. Down on the starter, not so easy. But chances are, voltage is not getting to it.
So we look for the STA RLY (Starter Relay?). Notice the Brn/Yel wire coming off the bottom right? I'm willing to bet that goes to the clutch safety switch. If the clutch is not pushed in, no ground to this. But I also recall on YJ's where there is a fuse that you can pull that will allow you to crank the engine with the clutch out. Used to hill starts in gear. I think we would need to see this part of the schematic to know how all that is wired.
Ignition switches do go bad but 'usually' it's kind of a gradual process....you can pretty much tell it's acting up....the most common is that it will crank and as soon as it starts and you let go, the engine stops....but it you hold the ign in just the right postion, it will keep running.
Good luck...