Coolster 125cc ATV - Will not electric start

[Coolster125]

Hello,
I recently picked up a used Coolster 125cc ATV that the previous owner had said they had push-started it for the past few years because the electric start didn't work. It did not have a battery when I got it. I have since installed a battery and a new electric start switch. However, nothing happens when I turn the key on and try to start it. I am pressing the brakes but still nothing happens. I should also note that if I try to turn the lights on, they do not come on, so it seems something is wrong with the wiring/system. Does anyone have a complete wiring diagram showing all of the connections? I have noted that there is one loose wire I am unsure where it is supposed to connect to. See the attached image.

I should also note that I can start the machine using a reverse drill or by bridging across the solenoid with a screwdriver. And the engine cutoff switch does not cut off the engine when running. I have to apply the choke to turn the machine off.

Any ideas? Thanks for reading.