I’d be happy to see the FBI take that role on for federal assignments _if_ that reduced budgets and the current rot was excised in some meaningful way with mechanisms put in place to hold individuals accountable.
(Actually, I don't know about any country where it's completely civilian up to now. But that's not something I ever looked.)
Seinfeld - Newman interrogates Jerry--Mail fraud : https://www.youtube.com/watch?v=jDmA2xoIoFU
There definitely needs to be cuts in 3 letter federal agencies. They're out of control.
In the end though, it doesn't matter because eventually over time all of those agencies with too much power will become corrupt. Massive oversight and constant cleaning out of people that have been entrenched too long is necessary.