I'm not sure if this is the point you're making, but I think the problem starts with the United States having a lot of guns in general, not just in the hands of police. The United States in the only country in the world with more guns than people.
I'm not saying US police aren't too trigger-happy, but comparing gun use by the police in the US to other developed countries that have a fraction of the guns per capita (in England, for example, there are .04 guns per capita), isn't apples-to-apples.
It's guns in general, exactly as you suggest. It turns even simple encounters like a traffic stop into potential life-or-death situations.
This is not the only issue with American policing (by far), but if the starting assumption is that people on both sides of encounters are in fear for their lives it's not surprising that we see a lot of bad decisions.
I'm fairly pro-gun but even I'll concede this is a big BIG reason why police shoot first.
A bunch of US police studies found that if the suspect looks like s/he is pulling a gun, then you pull first and shoot first. And this being the US, that's not a crazy or unlikely situation (compared to somewhere like Japan or the UK).
Generally the studies backed that the winner of the gunfight is 1) who pulls a gun first, 2) who shoots first, 3) who scores a disabling hit within the first three (3) shots.
The downside of teaching all of your officers this means that they see someone sneeze or move their hand incorrectly and they assume s/he's reaching for it and goes for a killshot with 3+ rounds. The drunk (white) guy who got shot by the cop when he tried to pull his pants come to mind[1].
I'm not saying US police aren't too trigger-happy, but comparing gun use by the police in the US to other developed countries that have a fraction of the guns per capita (in England, for example, there are .04 guns per capita), isn't apples-to-apples.