My guess would be the excitation capacitor. It needs to be able to store a small charge, which in turn will maintain the magnetic field in the exciter winding. This in turn causes the spinning rotor to induce voltage into the output windings. I had to change the capacitor on an old Onan generator for the same reason- failed capacitor would not excite the field. You could try connecting a small battery across the cap. for a minute, then test it for voltage with a voltmeter to see if it can still store a charge. Also, if it looks like an old condenser from a points ignition, you could substitute one of those and see if it works- the one on the onan looked just like a condenser.
1985 Tri-Z 250
1985 ATC250R