The idea is simply to avoid conflict of interest, and as you know and described, that is inherent when the executive investigates himself through one of his people. We do the best we can.
Unless someone comes up with something better, I think the "best we can do" is remove the prima facia conflict of interest by having the executive branch officers investigated by appointed officers of the House.
This Trump thing also make it important IMO to clearly define the steps required to bring about valid impeachment (formal charges), or else we get prima facia conflict of interest and "guilt by accusation" as the Lying Left is trying to do here.
Nobody is standing between the House and senate doing whatever the heck they want to on this. They have been investigating, and Congress is one of the primary sources of accusation. See Schiff, Warner and a handful of others. They don;t have to step aside for Mueller either. They can tell him to go to hell, we have this.
What Congress can;t do is bring criminal charges. That's the line. But they can do all the investigation their little hearts desire. The regulation of that is election, fifth amendment, fourth amendment.
-- This Trump thing also make it important IMO to clearly define the steps required to bring about valid impeachment (formal charges), or else we get prima facia conflict of interest and "guilt by accusation" as the Lying Left is trying to do here. --
The lying left is able to defeat any system. it is a waste of time and effort to constrain them with statutes, regulations, and rules. They can only be defeated with ridicule. They are not reasonable, they are irrational and outcome driven.