American exceptionalism is an extraordinary claim and needs extraordinary evidence to support it. It is far more likely that that our cops have a third-world standard of aggression than that intimidation and violence are that much more necessary when dealing with Americans.

I in no way said that aggression and intimidation are more necessary when dealing with Americans. I also find it baffling that anyone would feel it is an extraordinary claim that America has differences from other countries. It also seems obvious to me on the face of it that those differences impact in practical terms what is logistically effective.

Some differences I know of off the top of my head:

America is a relatively young country for a world power. It is much, much younger than many European, Asian and Middle Eastern countries.

There is much lower population density in the US than in many other developed countries. It is also huge compared to many other countries. The combibation of low population density and lots of land means, no, you can't just let someone go and then hunt them down later as the OP suggested. They absolutely can disappear on you and can then potentially go to another state or even another country, such as Mexico or Canada.

Unlike Europe or Japan, large parts of America still have large, very dangerous wild animals like bears and moose. Foreigners often decry America as some land of savages for our right to bear arms, something you cannot legally do in some countries, and our propensity for exercising that right. I don't own a gun and never have, but I assure you that if I moved to Alaska I sure as hell would acquire a shotgun and I would store it the way my father stored his shotguns: unloaded but not locked up, with ammo readily available. If a bear gets into your house, you do not have time to unlock a gun case.

Given that we do have a right to bear arms, the police absolutely are dealing with a more dangerous situation here than the police deal with in some countries. That increased danger makes their job riskier and it means failure to contain a threat can have terrible consequences.

I am sure there are a great many other differences, but those seem to be a few very salient ones. If you want a longer list, I imagine you can readily find more with a quick google.

On a sidenote: All the points you listed apply to Canada pretty exactly and they hardly need police at all (in the parts that i know very well which is BC and NWT).

Also thinking a shotgun is gonna help you with a grizzly is a major fallacy. What you need is bear spray (most effective), or a .45 magnum and exceptional aim to hit the brain on shot #1. If you shotgun the bear hes gonna live for 3-4 minutes and those will be your last (I encountered a couple of grizzlys in the wild, one fake charged me even from under 20 yards).

We expect our schools to perform as well as European schools. In some places, that actually happens, and we should expect no less. Our police should perform as well, too, especially for what we pay.