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.
The point of our disagreement hinges on the differences between impeachment (judgment limited to removal and bar from office) and criminal indictment. It's best we not mingle those terms as though they are equivalent. They are parallel in many ways (accusation, trial, judgment on conviction), but different by the entities involved in accusation and trail, and on the penalty upon conviction.
The House doesn't indict. It can't. It can only impeach and with the concurrence of 2/3rds of the Senate, remove from office. I think vital to have a coherent discussion, your statement would have been more accurate as "Since impeachment of an Officer may ONLY be made by The House, it seems reasonable that the things required to bring impeachment including investigation would also belong to the House."
If the executive wants to indict, the system you've set up puts him at at least a timing disadvantage. The executive (the entity with the power to indict) has no investigation. The House could of course share or hand over its impeachment investigation, but the Congress lacks the power to bring criminal charges - for good reason.