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.