EUREKA! The clue was actually in that last screen shot above: at the end of the first menu, it says “World Override: GameMode: Not overridden!” Which was obviously not what I intended. When I tried to pick “None” from that menu, it just went back to “Not overridden!”
Apparently “None” is not a valid choice for a game mode, and when you pick that, you’re actually saying not to override the project default at all. I chose “GameModeBase” instead, and now when I run, I no longer see my HUD.
I’m still a bit confused because the World Outliner still shows a ton of auto-created objects, though. When GameModeBase is chosen, it won’t let me select any options for the default pawn class, etc., but apparently creates them anyway (?). Possibly to create a truly empty level, I would have to define my own GameModeBase subclass that refrains from creating all this extra stuff? But that sounds too advanced for me, so I guess I’ll just live with it for now.