I'm not sure how much this might help, as it looks like some of the wiring is hacked up, but...The Yel/Red wire supplies power to the solenoid from the START button when it is pressed. There is a Grn/Red wire on the other terminal of the solenoid that (on the "F" bikes) went to a safety circuit that required the clutch to be pulled in, or the engine to be in Neutral (or both). The diode that comprises this 'safety' circuit ground is that black thingie that is plugged into that rubber plug that is mounted on the bracket at the bottom right of the 1st photo you have above (first post). You'll find the Grn/Red wire on one side of that mounted plug (the diode part unplugs from it, it's a separate part).
If the safety circuit has been removed/changed then that diode thingie may not be working anymore. In Honda's earlier bike, the Grn/Red wire was just a Green (ground) wire, and you didn't have to pull in the clutch or find Neutral to make the starter work. Apparently, Honda thought that was dangerous enough to make you either find Neutral yourself (not handy in traffic) or pull in the clutch (instead) to enable the START button to work.
All this said: the START button in these bikes has 3 wires tied to it: one is the Yel/Red you already know and love, one is a Black (Power) wire (from inside the headlight bucket), and the 3rd one powers the headlight from the Black wire when the START button is NOT pressed (this make the headlight turn OFF while starting with the button). If the PO installed a high-powered headlamp into the headlight, that melted the START switch inside the handlebar, so then the Yel/Red wire doesn't get any power when the START button gets pressed. So, it sounds like he tried to make some other sort of START button up after that?
You can get new handlebar switches, from places like PartsNmore.com or their USA (Florida) branch at VintageCB750.com. They work well, I just installed some on a CB550K1.