Onan 5000 Emerald Plus?

This model has two spark plugs. Gap at .25.
Also likely the carburetor is gummed up from sitting. Clean carb bowl and retry.

Push start and it will crank and run but let go of the switch and it shuts off

These units use LATCHING RELAYS to hold in ignition and fuel pump signal power and if the relays (multiple) fail to transmit a hold signal to the main power relay your generator will act exactly as described. You over-ride the latching failure by holding in the button but unfortunately, you are also keeping the starter motor engaged.

Check the generator's umbilical harness connector both sides for loose or corroded terminals. A malfunction here will affect both the generator's controls and the remote switch.

GOOGLE "Latching Relay"

And CAIG INDUSTRIES DE OXIT red D100 contact cleaner has resolved about HALF of these malfunctions for me. Available on AMAZON and priceless. Disconnect harness junction put a few drops of cleaner into each side of the connector halves and reconnect. No other brand of contact cleaner will work. Do it to the inside remote start connections too.

All good answers. Carb needs cleaning for sure.
There’s also a bundle of grounds screwed to the frame. Some are control grounds so find then and clean them and the frame.
For what it’s worth I like this type of generator I’ve had several. You’ll have to learn about it to take care of it. Do you have the service manual ?

I don't have the service manual yet.

You’ll need your a model number and spec letter at the end of it to find your manual here.
What is your model and spec ?http://www.skilledcrafting.com/onanfiles/
This should help you find your manuals.
https://www.dropbox.com/s/o6sxv50bbbvp3qb/932-0112%20Manual%20Index%20.pdf?dl=0

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)