I wasn't as precise as you but my claim was correct, i.e. Captain stays where he is. That the card he copies also stays where it is is less relevant, except for piling and covering up (for example if Encampment would get set aside, you could pull off tricks like getting at Plunders) issues.
What you mainly care about is that you kill off your BoM if you play it as Death Cart whereas your Captain survives if you play it as Death Cart. Whether the Death Cart in the Supply that you copied gets trashed or not is of secondary relevance.
You are incorrect. The "leaving it there" phrase does not apply to Captain; it applies to the card Captain plays. It says that the card Captain plays does not get moved into play.
It is true that Captain stays where he is, but that's simply because
that's what cards do. If nothing says to trash it or return it to a supply pile, then it stays in play. When you use Captain to play Death Cart, nothing says to trash Captain, so nothing happens to Captain.
You are possibly missing that you do not play Captain
as the other card (the way you do with BoM). Captain says "Play a non-Duration Action card from the Supply..." It does not say "Play Captain
as " the card. Captain does not
become the card.
You focus on mechanical details, I focus on practical implications and comparisons with existing cards. In my opinion both perspectives are useful but you seemingly think otherwise.
If you want to compare to existing cards, you should be comparing with Necromancer, not BoM.