I have had this remote start problem with two different Onan gensets. Both times the problem was with the remote switch wiring, not the switch. This wiring runs to a "molex" type connector that plugs in to the side of the genset switch housing.
First, look for corrosion of the pins in the connector (squeeze the ears and pull gently to unplug) and clean as required; also while the connector is disconnected, ensure that all the wires are properly crimped to the pins in the connector. Correct any issues if found and then reconnect and try to start remotely.
Second, both of my gensets were supplied with only about 12" of wire on the connector and the OEM supplied the wiring to the dash. These wires were spliced using crimp-type connectors which were hidden inside the plastic wire loom. Find the splices and inspect carefully; this circuit usually works by connecting the start circuit all the way from the gen set to the dash and back again so it MUST be a low resistance connection. One (or more)is very likely corroded. Re-splice if necessary and try to start remotely.
I hope this helps.