I think this is true in some circumstances but I think people are usually leaving the "situation" (like 90% of the time in my experience). I don't think the statement should be used anymore for this reason.
People are, indeed, almost always leaving the situation.
But at least in my experience, it's still the case in the past few years that every time someone has told me they were "quitting this job", versus "excited about this new job", it was specifically about their direct manager's effects on the situation.
(Sample sizes for any individual small, of course.)
Either way, I still think the saying is useful for intentionally reminding people of how much influence your direct manager can have on your work experience, because I've found a lot of people, particularly new hires, don't appreciate how much your experience can vary across managers.
