This is interesting. In my country (Poland) parliament members have legal, immunity from criminal prosecution and arrest to the point of police not being able to stop them if they drunk drive. There have been some abuses like that.

The law is such that a prosecutor that wants to prosecute them has to ask the parliament. Then there is a vote and the parliament decides if the immunity is taken off.

In a healthy democracy, where there are more than same two parties switching the rule to one or the other it is very likely the current opposition will be the majority next time and they will vote to strip immunity from those that try to use it as a shield for criminals.

I think the price we pay for this (delay in getting justice) is well worth paying so the justice system can't be used as a weapon against political opponents easily.

The rules and laws allowing the federal government to take over a state case against a federal agent seem much more damaging.

The cops involved in the most recent Minneapolis shooting will almost certainly face no repercussions because of this. The state can bring a case but the feds are clearly uninterested, they would simply take the case into federal court and spike it.