The Independent Counsel Prosecutors had that power, without need to report to the AG, but I believe that position of Independent prosecutor/counsel was let to expire in 1999....after that, it has been Special Counsels/prosecutors, which has less power than the Independent Counsel had....they do need to update the AG or the Asst or Deputy AG and FOLLOW the guidelines in their hire, and scope of investigating preauthorized. They can be fired by the AG, but not without cause or without a replacement...is my understanding...but not 100% certain.
The Senate can't confirm an executive branch position without a president nominating them....so what you are saying is basically, The President needs to pick the independent Special Counsel who is being hired to investigate the President's crimes or in this case, The president in power who is the opponent and enemy of the previous president, has to nominate the Special Counsel who will be investigating his past opponent and former president, and future opponent, in order for this position of Special Councel, needing Senate's approval....
That would be highly charged and unfair in the eyes of Americans in my opinion and that's why I do not support the stance that this position is one which can be granted only by a President nominating and the Senate to confirm...