Have you checked the fuel pump yet? The sitting/refusing to start for no reason/dying randomly symptoms seem like that might be your culprit.
Pull up the rear seat and unscrew the cover in the middle. The pump is right under there. Next time it refuses to start, turn the key to "on" but don't start the engine. Listen to see if the pump is running to prime the fuel system. Also inspect the wires/contacts to see if it looks like they're overheating or corroded.
Is the check engine light on? If so, what codes is it throwing?
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