Generally, the only valid federal act is that which is authorized by the Constitution. Generally, if it is not in the Constitution it is not a valid federal act. As far as impeachment, the Constitution is silent about how to arrive at impeachment proceedings but it does say the House has sole authority to impeach. "We've always done it this way", is not necessarily the standard for validity.
Although it makes sense for the DOJ to investigate federal officers in other branches because there would not be prima facia conflict of interest, the Constitution certainly doesn't say the DOJ which is in the executive branch and under the President, should investigate the President or executive branch officers.
First of all, what legitimately triggers impeachment (charges)? I surmise probable cause (PC). But by what means is PC determined? Not sure. In regular police work, usually there reasonable suspicion to launch an investigation that may lead to PC and custodial interrogation or arrest. If the suspect was a federal officer in the executive branch, the only non-self-conflicting branch to do all this stuff would seem to be the House, I suppose under the heading of their "sole power" to impeach. The Senate would be the trying court so I don't think it would be the Senate. Not the Supreme Court. They don't investigate.
I think we have an open issue of who should investigate reasonable suspicion or PC with the President and possibly other federal officers in the executive branch. I think constitutionally and logically that task would probably fall to the House. Of course potential political shenanigans abound there too, but at least there's not prima facia conflict of interest.
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.
My view of the impeach vs. criminal charge issue is that the question is academic. If there was evidence that the president murdered a judge, the crisis gets managed somehow and the details aren't all that important. I think the correct approach is impeach and remove him first, then indict him. But I don't think it matters if the order was reversed. The difference amounts to a certain amount of shuckin' and jivin' in motions before the judge.
Impeachment itself is a political question. It does not follow the usual spectrum from articulable suspicion to probable cause to (whatever the standard is for indictment) to beyond a reasonable doubt.
But your reference to the standards of evidence is interesting, because from what I can tell, there is ZERO articulable suspicion (the lowest level) that the Trump campaign was involved in campaign coordination with anybody in Russia. The suspicion is "from the imagination," taking disjointed facts, and coupling them on the basis that Trump won the election. If he had lost, the investigation would not exist. OTOH, if there was a crime, and he lost, the investigation would persist.