The government has a monopoly on force. That force should be weilded by the fairest and most impartial people possible. Police, investigation agencies, etc., should be as free from bias as possible.

Now, you have multiple ways to get to that point, and people have different opinions on the purest way to achieve this. But, electing them doesn’t seem to be the way. Tyranny of the majority is too strong. And appointment by elected officials is equally problematic. So how then does a system establish that is not subject to abuse by those with power?

I would argue that the best system for appointing law enforcement seems to be via a benevolent dictator or monarch or their representatives. And it only works for their lifetime, unless the inertia of the benevolent institution can be sustained. Well, it’s a crapshoot but stable at least for the lifetime of the monarch or similar.

I’d also entertain citizen lotteries for these sorts of positions. But that’s a crapshoot on shorter timeframes.