2002 Cadillac Deville with 78,000 miles is in pristine condition! On rare occasions, albeit never conveniently, the car will not start. No clicking or clacking like a dead battery. All lights work indicating the battery is good. Radio also works. When this happens I turn the key back to the off position, wait a minute and try to restart the car. And somehow the car restarts.
There is a relay the drives the starter when you turn the key to “start” That relay is probably starting to fail. Easy to replace. It is in the fuse box underhood located next to the engine.
It is #36 and labeled “Starter 1” See here; https://www.autogenius.info/cadillac-deville-2002-fuse-box-diagram/
Install a new relay and see if that solves it. A relay is cheap. Hopefully it is not the starter itself, that is a headache to change.
The next time the engine won’t start, with the dash lights on, step on the brake pedal and shift the transmission into neutral and then try starting the engine.
If the engine starts, it points to a problem with the park/neutral safety switch.