Your generator likely uses the chassis battery and may have a weak connection or failing starting battery. The main engine alternator charges the chassis battery when running, explaining why the generator starts after running the engine.
On 1991 E350, battery ground typically terminates at engine block near starter or frame rail under battery. The fender tie is likely a routing point - continue tracing to find actual ground bolt. Clean to bare metal and use proper gauge cable for replacement.
Starter relay is in engine bay fuse box on Ford 7.3 Godzilla. Starter staying engaged usually means stuck relay contacts. Replace relay - should be warranty covered on 2022 unit.
Ignition lock cylinder gear failure fixed by replacing lock cylinder assembly. Remove steering column covers to access. Consider locksmith for complex repairs.