So now that I have more info on what happened and have viewed the video I would rule out any issue with fuses or relays. The car starts and runs. So your getting fuel and spark.
From what you described and what I see in the video (the smoke). The issue is not transmission related it is engine.
From the smoke at the tail pipe it looks like your getting oil into multiple cylinders fouling out your plugs.
Draining the oil and looking at it and checking the oil filter are not bad ideas.
Any information you can get helps in diagnosing the issue.
But if you pull the plugs you can tell from the condition of them lots about what is going on.
For example.
If your problem is a lifter you will see a fouled plug on that cyl. the others will be good.
If you have something going on like a total failure of the AOS all the plugs will show oil fouling.
If your issues is say a chain tensioner pad and it has caused the cam to change timing your plugs will show it on that bank only.
If your issue is something like a failed AOS It could cause all your symptoms. That car will run like crap because the plugs are fouling out. No power, the engine will rattle and knock all kinds of things.
So I would for sure pull the plugs, if both banks show signs of oil fouling then I would look at pulling the throttle body and checking the intake for oil.
If only one bank shows oil fouling then that probably points to some other issue.
anyway pulling the plugs and checking there condition can tell you lots about what happened and the overall condition of the engine.
They will tell you if the issue is on 1 cyl. or multiple cyl's. 1 bank or both banks..
My thought on the Relay is that it was just a fluke. car will probably start with the old relay.
Also you said when you tried to restart the car out on the road that it seemed like it had no compression. So once all the plugs are out and checked you can do a compression test on all cyl's and get more info on engine condition.
Based on what you have said about what happened and what I see in a short video the above is how I would proceed.
Keep us posted I am always interested in outcomes
