That is not always correct, the starter uses a different ground path than the rest of the system.If the starter turns over and the engine fires the fuel solenoid is as grounded as can be.
I suggest there's a faulty ignition switch from wear or corrosion.
btw, starter solenoids have pull and hold coils but few other do. The fuel solenoid is so much smaller and requires far less current to hold it. It likely doesn't use a relay either.
The fuel solenoid may be more like a diverter valve solenoid that draws a few amps vs a starter that takes nearly all a battery has.
My original thought was a faulty key switch, that I replaced, after determining that it was not a fuel problem. The new switch did not solve the problem. Do you think it could be a bad solenoid or something else?? I am not very educated on these mechanical issues, so all of your help (in laymen's terms) is deeply appreciated. Thank you very muchThe starter and fuel solenoid are grounded through the engine block. If they both work in the key start position they are grounded ok. Neither has a ground wire/connection to fail except the main ground to the engine. (wouldn't crank over)
A bad key switch might not send power to the solenoid in the run position I've replaced two ign switches and problems ceased. One was in '98 and one was in a '04. Both had spent plenty of time parked outdoors, but I suspect contact wear due to age & hours.