Two things:

1) Some quads have AC powered lights (headlights and tail lights, but *not* the brake light). How did this work before? When you turned on the ignition switch and turned on the headlights (but leaving the engine stopped), did the headlight come on?

If so then your lights are DC powered. Back when your quad was working, if the headlights only came on after the engine was started then your lights are powered off raw AC voltage directly from the stator. Note how I specified the "back when your quad was working" part because it is important.

2) When you jump the solenoid and start up the engine - even if your lights are DC powered - the running engine will make 12 volts through the regulator and power up the 12 volt bus. The headlights will work.

It sounds to me like you have a break in the 12 volt battery path through the fuse, ignition switch, and brake starter safety interlock wiring. Once the quad starts the regulator feeds the other side of that break - allowing the stuff on the 12 volt power bus to work.

 

Maybe aromao got it fixed and is using it to drive to and from work everyday.