Obviously a Principal Investigator (PI) has many responsibilities toward his or her subordinates, including mentoring them, helping to direct their research, providing funding, and resolving difficulties.
But, in practice, it is often possible for a PI to avoid many of these responsibilities and "get away with it". By this I mean that failing to address these responsibilities has little effect on the PI's own career.
For example, it is often understood that a PI should have some input when PhD students are writing manuscripts. A PI with a large lab may have a policy of leaving students to fend for themselves in this regard. With some intelligent selection at the hiring stage, this PI may ensure that most of his or her students are already capable of writing papers with no assistance. The few that slip through will fail to have a productive PhD, but this will not directly impact the PIs career all that much.
Similarly, a PI may be unable to come up with good research ideas, but this can be compensated for by actively seeking out graduate students who do have good ideas. Since it is rare to explicitly state in science who had the idea for a project, the PIs CV will still look good thanks to the papers the students end up publishing.
However, an example of an obligation is applying for grants. If a PI is ineffective at securing grants, it is very unlikely that their career will not suffer. Students and junior lab personnel simply don't have the knowledge or experience needed to put together a good application, and while post-docs can secure their own funding, the majority of funds in a typical lab is brought in by the PI. If the PI decides to not bother himself with obtaining funding, the lab will become financially hamstrung, research will suffer, and even to a casual outside observer it will be obvious that the scientific output of the research group is appreciably constrained.
My question is, which responsibilities cannot be avoided by a PI without necessarily producing severe negative consequences? If after a cursory examination, a PI's career seems to be going well, which responsibilities can we be certain have not been neglected?