Your clarification makes sense - a PhD advisor with a bit of NSF or other grant money would likely only fund a grad student to work on a problem that has a reasonable likelihood of generating some concrete, positive, and publishable results. This typically means an incremental result on a problem of interest to the research community.

That having been said, an accepted proof that P != NP would result in a Turing award and an additional $1M for solving one of the seven Millenium Prizes. This has been an open problem for decades, and it is a problem of enormous importance and visibility.

Although that might be based on the perceived likelihood of success. After all, pre-Wiles, I doubt there was much chance of a grad student getting funding to prove Fermat. Nor is a grad student likely to get funded to prove the Riemann Hypothesis.

Because (I'd guess, as a PhD student in another CS field) any advisor worth its salt would advise a grad student to work on something else first, get tenure, and then maybe approach this problem. Until yesterday, most researchers agreed that the problem was unapproachable. Nowadays being a researcher is a job that requires steady progress, so you must focus on approachable problems. For P vs NP there are tons of results on classes of techniques that _cannot_ work—you'd have to learn those first to make a serious attempt.