A brush type gen head has an exciter winding with a diode, an AVR (Automatic Voltage Regulator), and a set of brushes on slip rings to supply the power to the field. The best thing to do is put a voltmeter on the output and try to start one of these motors that won't start and see if you are getting a huge voltage drop. If the voltage is dropping out of spec (anything below 110V) during what should be the starting surge either the exciter winding, the field winding or the brushes/slip rings are highly suspect.
I think the AVR itself is OK because you are getting normally regulated voltage at no load. But the AVR can't supply the field with more current if the exciter has gone bad**, nor can it supply the field with more current if the slip rings are dirty or the brushes worn badly. And if the field has a shorted winding in it due the insulation in the winding breaking down, the field will be weak and limit the generator's surge capacity.
**some gensets do not have a separate exciter winding, but instead take power from one of the stator windings and run it thru a diode for the AVR current. If your gen head is this type, then you could have a bad stator winding also.