There are numerous causes of symptoms you described. As OP's have noted, the "run " relay on control board have to latch in to power ignition and fuel pump. As you crank over with starter, the ignition and fuel pump are directly powered. The run relay on control board has to "see" 120 volts being produced to energize. So most common cause of shutting off when you release starter switch is 120volts not being produced by gen.
Common causes: rotor field brushes stuck or slip rings corroded/dirty, voltage regulator failed (very common), control board not "flashing" field when cranking, or...worst case ...open or shorted winding in rotor or stator. Bad LOP switch or low oil pressure will cause this problem, but sounds like you covered that. The field brushes are located in stator housing behind air cleaner (have to remove air filter housing to access)