SPOILERS
The Irish? (Scottish?) writer-director seems under the impression that police violence is so epidemic and unprosecuted in America that a rural Missouri deputy could without any justification other than anger brutally beat the advertising guy and fling him out of a second story window, in front of witnesses including an outside law enforcement authority sent to said town, no less, and not be charged with felonious assault if not attempted murder.
Another issue I had is Woody Harrelson's sheriff character was smart enough to have not left Sam Rockwell's deputy the goodbye message he did without foreseeing the deputy would go apeshit and try to hurt or kill somebody. The sheriff would've been more careful to somehow head that off from happening. And he let other misunderstandings exist in his goodbyes; he didn't make it clear enough it was the cancer and not Frances McDormand's character's campaign that was the reason he was choosing to die right then. IRL the sheriff would have known to do more to keep her from being blamed.
I still enjoyed both movies overall but had those and similar implausibility complaints with Three Billboards and In Bruges as well. Probably a european thing but the auteur doesn't understand firearms, for one thing. 100 million Americans do, though, and we might watch your movies. (Admittedly, the same ignorance doesn't worry, say, The Walking Dead producers either.)
If I'm investing my time in a drama, unless it's meant to be surreal or a dream, I appreciate it being realistic.