Sounds like it might be an electrical issue (battery,grounding,etc), or maybe a dirty throttle body...but it's weird for sure.
I am not sure about the oil light, I would think oil starvation would shut the engine off for good, not so intermittently...I assume you've checked your oil levels regularly?
Are you getting any immobilizer DTC faults? I am wondering if a bad speed sensor could be the culprit...it's not unheard of.
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