I find it interesting that Vader is referred to as Tarkin’s “henchman.”
You certainly sense in the movie that Tarkin has some authority to restrain Vader, but hardly that he is in charge.
Are you sure you’re not letting TESB+ material colour your view of Vader’s status in SW?
Perhaps you’re right. It’s been a while since I’ve watched Star Wars in isolation. I usually watch them once a year in rapid succession which might make me not appreciate these distinctions.