What is the defining characteristic of post-apocalyptic American science fiction? The thing that, by the creator's measure, marks it as a fallen world? I'd argue that it's a lack of police.
It's the transformation of the entire world into the frontier, or rather, the frontier as viewed by settler-colonists: the few civilized towns, safe havens primarily distinguished by the presence of police, beset on all sides by unreasoning killers that spawn from the desolate wilderness. In real life, it was the continent's indigenous population that was portrayed this way; in American post-apocalyptic science fiction, it's drug-addled raiders, cannibals, mutants, inferior people with inferior weapons who produce nothing of value, but nonetheless pose a threat by their sheer numbers and utter depravity. Or perhaps the story takes place in a crumbling ruin of a city, where the fear of lawlessness manifests in a vision of street gangs as imagined by anyone ignorant enough to believe that old chain email that claims "gangs initiate new members by driving with headlights off at night, then killing anyone who flashes them."
You know Fallout raiders? The guys who kill and rob because they're bad and evil, and mindlessly charge into your gunfire until they're all dead, without ever considering retreat or surrender? That's the basic idea.
It's a conservative, Hobbesian view of human nature, and so it's right at home in American culture. It's also a way to avoid grappling with the reality that fascism is capitalism in decay - that as conditions in a capitalist society like America deteriorate, the police and military won't vanish but will instead only get more cruel and vicious. After all, as people get worse off, they will get more desperate and violent, creating the excuse for ever-increasing state repression, an ever-intensifying reaping of the poor for the sake of the rich. Those on top will spill oceans of blood to ensure that even as the pie shrinks, their piece doesn't. In the end, that's your best candidate for raiders: the hollow shell of a state so degraded and honed for violence that it's incapable of doing anything but killing and robbing the people in its borders, carried out by those with vast stockpiles of guns, ammunition, armored vehicles, and fuel. Once supply chains collapse to the point that the state can't even maintain its armed forces, it's not hard to imagine things going full Thirty Years War, with roving gangs of soldiers and cops sustaining themselves by hopping from town to town like locusts, stripping each bare before moving on to the next.
Obviously, this doesn't apply to all American post-apocalyptic science fiction, but it's common enough that I think it warrants discussion.
As you point out, much of post-apocalypse fiction (and, really, a lot of our science fiction as well...) lifts its themes and tropes directly from "Western Frontier" fiction. I don't think there was ever really a point where any of those themes got examined critically...? I've heard that after WW2, from the 1950 through to the mid 70s, there were only three national television stations -- but there were around 50 "wild west" television shows in active broadcast on practically any given day. From the network's perspective, it was easy money. Simple props and costuming, exterior locations were either the California desert or your buddy's private ranch. Good guy, bad guy, tense shootout; it writes itself!
The story I heard is that Blazing Saddles killed that cash-cow. One fairly silly movie that asked the fairly simple question; "Could a black man be the sheriff?" was enough to end the trend. Seems like the television studios were presented with an option: keep making historical fiction, but maybe actually think and ask questions about the era you're purportedly writing in --
or
-- find a new formula.The studios took the easy option (goodness, there are a lot of police procedurals...). Mass media never really explored the era it wore as cheep costume -- and now those shallow historical fictions are haunting us in our future narratives.
Echoing what you said and adding in Space and Vietnam changed the zeitgeist in the 50s/60s, and Westerns fell out of favor. Source: took a class on Western movies back in like 2008