How and why has the American cowboy fashion/culture persisted for so long up until our current decade? Every decade has a fashion and culture that later we come to our senses about. But not the cowboy apparently.
Simple: we still have cowboys, therefore we still have their clothing. As of 2016, over a million Americans were employed in farming and ranching, and there were more than 750,000 cattle ranches in the USA.
"Cowboy fashion" is entirely pragmatic. The jeans, overalls, hats, wide belts, boots, handkerchiefs, etc., all served a variety of practical functions for people on a ranch, and continue to do so today.
Many of them are also practical for other rural inhabitants, whether they work on a ranch or not. When I was living in rural Texas, for example, I had both a hat (because it was good for both rain and sun), and cowboy boots (because they protect your lower legs and are easy to clean).
Frederick Jackson Turner's Frontier Thesis regarded westward pioneering as a foundational characteristic of the United States. This view glorifies the olden days of the Western frontier, or Wild West, rather than cow husbandry in general: East Coast dairy farms, not involving pioneering, are not so romanticized.
Cattle was a common industry on the mostly arid frontier, requiring little infrastructure. Cowboys have an enduring cultural appeal in the U.S. because they are remembered as central to stories of the Wild West, still one of the country's core narratives.
A comment below has correctly pointed out that the dominant historical narrative tends to erase communities of Mestizo, Black, and Native cowboys in favor of Anglo ones.