The fact that the dashboard lights do not come on when the engine dies suggests to me that this is clearly an electrical problem somewhere in the circuitry that’s supposed to be enabled when the key is in the “ON” position. I’m inclined to suspect the ignition switch (key cylinder assembly) itself. Id want to start by checking for voltage at the relay that’s energized whose contacts enable the necessary circuits to see if voltage has been lost there.
You may want to try a shop that specializes in automotive electrical circuits. They’ll have the expertise to trace the circuits via the scematics and track down the intermittant connection or determine which relay is likely to cause these symptoms.
But I’ve recently been renminded that it’s a good idea to check the fuses first. You might get lucky.