That isn't contentious. What is contentious is your position that BEFORE CONVICTION and REMOVAL, the Officer is immune from executive-initiated action - even from being investigated on suspicion of criminal conduct.
Since indictment (impeachment) of an Officer may ONLY be made by The House, it seems reasonable that the things required to bring indictment including investigation would also belong to the House. It doesn’t make sense for one completely separate body to go on a investigation spree if they have no power to indict.
It would also appear that is not the constitutional intent. For whatever reason, the Constitution removed criminal procedure for an Officer from the executive branch. Maybe because it would be harder and less likely some personal frivolity or grudge, if done in Congress.