Random Trouble while Starting
As everyone knows who has a C5, small electrical gremlins are endemic. The car is wonderful, but it can keep you busy chasing them. As has been said many times, grounds are the source of some of them. But not always.
I just cured a problem that sounds like yours. About six months ago my 2002 Automatic Coupe began to occasionally not do anything when the key is turned. Nothing. The fan would run, radio would play, lights on the dash, but turn the key and nothing would happen. I started thinking it was the neutral/safety switch being defective. So I would try to move the selector from Park to Neutral repeatedly. After doing that ten or twenty times, it would start and run perfectly as usual.
This would happen maybe every other day. It would always start perfectly when cold or slightly warm. But drive it till fully warmed up and it wouldn't engage the starter solenoid and energize the starter, until I repeated the transmission selector thing again.
I took it to my big Chevrolet Dealership and told them the story. Ended up leaving it for five days for them to try to diagnose this problem. At the end of my visit, they could not give me an answer, but no charge. Said they checked the neutral/safety switch and the ignition system. To make a long story short, over the last six months, I have left my car with them for two more visits of a week each. Always same answer. No answer.
Several weeks ago I met a terrific classic car mechanic (think Deusenberg and Packard) that goes to my church, and I related my story. His name is Terry Weiss of Terry's Repair in Tyler, TX. He said my problem with the dealership was that they are all well trained technicians, but not mechanics. He said, "If the problem doesn't show up on the On-board Diagnostic Systems or in symptoms in the manuals, they don't know how to diagnose a problem."
Terry asked me to bring my car and leave it for the day, and he would look at it. I dropped it off at his shop Monday morning and it was fixed by
Tuesday morning. What he did to diagnose the problem, was to take a jumper wire and run from the positive terminal of the battery, straight to the terminal on the starter. And it started immediately. That simple test eliminated all of the keys and switches and possible causes.
He found that the starter had a cracked field that opened when it got very hot. As it is situated next to the right exhaust, it gets plenty hot. Once it cooled, it made contact again. So he replaced the starter and it's fixed.
I learned a lesson there. Look for the simple causes. Trust old mechanics thoughts. I still love my Corvette even with it's peculiarities.