In my experience established PIs tend to delegate a healthy fraction of the responsibility for managing technical staff, mentoring students, and even acquiring funding to their postdocs. In turn, faculty search committees see this experience as crucial, and competition for academic jobs is strong enough (~100 applicants per position, last I heard) that I would imagine any new PI should have a fair amount of experience under their belt.
At the PhD level and above, academia seems to revolve around managing one's personal brand. Publishing a few good papers isn't enough, you have to be "THE guy/girl" in an in-demand specialty to get a job. Your observation that this requires a business/PR/advertising mentality in addition to the ability to do solid research is spot on, but I think the game changes long before professor-hood, and I think that anyone who successfully obtains an entry-level professorship has already demonstrated competence at playing it.
At the PhD level and above, academia seems to revolve around managing one's personal brand. Publishing a few good papers isn't enough, you have to be "THE guy/girl" in an in-demand specialty to get a job. Your observation that this requires a business/PR/advertising mentality in addition to the ability to do solid research is spot on, but I think the game changes long before professor-hood, and I think that anyone who successfully obtains an entry-level professorship has already demonstrated competence at playing it.