This is a Constitional procedure to fill a vacancy in a Branch of our Government with specific Constitutional rules to follow. The bottom line is the investigative role in the procedure falls to the Senate. Certainly not to the FBI unless you are ignorant of the principle of separation of powers.
The Senate tasks the FBI to run background checks on potential SCOTUS nominees, REDSTATEWARD, and Sen. Lisa Murkowski (for example) has endorsed the use of the bureau here, though maybe the Senator is ignorant of the principles of separation of powers.
But, as usual, you either miss or skirt past the main point here. If this were a normal nominating process held under a normal administration and not the current party over country fervor that has gripped the GOP, granting a lifetime appointment to a Supreme Court Justice who
may be an attempted rapist would be unconscionable. So in normal times, it would be on the Republicans to investigate the allegations and clear Kavanaugh of that possibility. If they could not do that, they would move on to a different candidate.
Instead, we have a party that is rushing to put a potential attempted rapist on the court before November elections, that isn't investigating these allegations seriously because they are afraid they may prove to be
true.
The fact that this is all distinctly
not normal is why you are so dedicated to the argument that this is all business as usual. It isn't, REDSTATEWARD. Not even close.