There's a lot of grease on it, but it seems to be okay just from appearance. Where does it go so that I can test continuity? I'm sorry, I'm not understanding what this one connects to
I think in your troubleshooting process, you may have created one or more additional problems. You have received a lot of good suggestions. Some of them might have led you down the wrong path. Address one issue at a time and make sure you solve it. You have to have enough electrical savvy to ignore those comments that don't make sense related to your problem.
It's been a long standing issue, and it used to be that it could be fixed by wiggling the wire harness going to the ignition switch or, in the rare occasion, honking the horn.
Lets get back to the original clue you provided. Wiggling the wire harness allowed you to start the bike. Was a broken wire or defective ignition found? I think you stated the white wire appeared to be melted/recessed in the switch. Was this fixed? What about the provided examples of poor crimps in the harness? Was this inspected? Sounds like you still have this problem.
Also, you stated honking the horn started your bike a few times. This is another clue that needs to be followed.
We need to know how you wired a toggle switch into the start circuit. The ignition switch provides power to several circuits, not just the starter relay. You may have caused another problem with this mod...