Yes I know that the FBI does BIs.
And Hillary got copies of pertinent BIs via her WH Security Chief (former bar bouncer) Craig Livingston! FBI leadership is at the top if not the ring leaders of the current scandal.
Possibly the only way your proposal would sort of work at least the way you are describing it. (and even then as you say nothing is foolproof!) Would be use a British style civil service system. Where you have very powerful civil service positions - permanent secretaries directly under the appointed politician\policy maker. (.. https://en.wikipedia.org/wiki/Permanent_secretary ....)
There they have the power (Should they chose to do so!) to thwart the politician. If the politician’s request/demand/action in the permanent secretaries mind damages the institution, his/her career, etc. (Their careers/promotions, etc. are completely outside the power of any politician!) So your BI vetting agency would have to be full of people like that, who could flip a bird to any politician that made a “unusual” request. Still , it gives such agency enormous power. The Rats would immediately try to unionize it!
We HAVE to do something to keep scumbags out of the body politic, and BIs are just one tool we could use.
I really don’t think setting up a huge new bureaucracy is the way to do it, but at the moment, other than using what is already in place for BIs, I cannot think of another way to do it.
Perhaps just the THREAT of having to undergo a BI would keep the riffraff out?