There was a fuse box on top of the battery originally. Somebody yanked it. I see in one of your pics, they added a couple in-line glass fuse holders instead, but it only looks like 2. There were 4 fuses originally. Looks like they eliminated 2 and just connected their wires together directly. Here's how an original fuse box was connected up .....
You need to start looking at some wiring diagrams. Above is an excerpt from one. Look at several because some show things better than others. Not all will have the fuse box labeled as well as the above pic.
You have 2 heavy wires running to the starter solenoid but there are 2 smaller wires as well. Those are the ones that trigger it when you push the starter button. The blue/white wire runs up to the button. It provides the ground for the circuit when you press the button. But, for it to make that ground, the left switch assembly will need to be in place. It provides the ground through the handlebars over to the right switch and the starter button. Also, it could be as simple as a bad, dirty, rusty connection of the big cable down on the starter motor. That's all that was wrong with mine. Starter didn't work when I got it, nothing, nada when you pushed the button. I cleaned that connection up and it's worked fine ever since.