Oh absolutely, 98% of superheroes, and the entre foundation of Civil War, are based on that.
But whilst citizens arrest and the like can be used as some sort of a foundation for a lot, cold blooded murder is not included.
As others have noted, it's the fact that some non-Bat person hasn't shot the Joker full of holes then run a steamroller over his head, that is the truly (if editorially mandated) problem (and why I would ever give the Joker the bodycount he does.
When he's focussed on committing crimes, culminating in an attempt to kill Batman and Robin, and there maybe a henchwork related fatality, so be it, but everybody and their mother in the DCU must have lost a loved on to the Joker by now.
I miss "capers", the old "series of themed crimes which don't revolve around mass murder" notion. We need more of those...