If I can modify that slightly: The staff engineer isn't accountable for people that have a rough patch or who are assholes (and the manager is). I've found as a staff engineer that I definitely have to care, adapt to the rough patch, and campaign/do politics to get the assholes out, but I'm not accountable for it.
(I originally wrote this with "responsible" in place of "accountable" but that was fuzzier; surely as a staff engineer I'm responsible for more than my direct contributions even if I'm not directly accountable for them.)
On the other hand people's sense of who they should really listen to always skews to their manager (the person that can fire and/or promote them, or at least has the most say in the process).