One of the premises of this video is wrong. The author claims that rockets are staged primarily because the engine bells need to be a different shape at differ

One of the premises of this video is wrong. The author claims that rockets are staged primarily because the engine bells need to be a different shape at different altitudes to maximize their efficiency. While it is true that engines designed for atmospheric and vacuum flight employ different bell shapes for the reasons the author describes, this is not why rockets are staged. The primary driver behind staging is to shed a portion of structural mass after it's no longer needed. For instance, there's no reason most rockets couldn't be designed to lift their payloads directly to orbit in one stage, but by the time the vehicle reaches the end of its flight profile, it would be accelerating a great deal of structural mass to orbital velocity before deploying the payload. The near empty fuel tanks, large engines needed to lift all that fuel you've since consumed, etc are all dead weight at that point. If instead the vehicle drops that mass mid flight and continues on to orbit in a much lighter configuration, the total possible deltaV of the system is greatly increased. Using second stage engine with a bell designed for vacuum only makes sense here, but that's not why rockets are staged.