As with most, if not all, topics popularized by cinema in relation to the Wild West, it's fiction. There's no basis in reality for it.

The type of scenario that you describe is referred to as a "town-tamer" Western. The origins of this sub-category of Western films can be traced to the renaissance Western (1939-1941) that revived the genre and made it a more lavished and well-promoted genre than in the past. The Second World War put a stop to this revival, but it returned in full force after the war. The town-tamer Western during this period is best exemplified with Dodge City (1939) and My Darling Clementine (1946). These two films practically established the blueprint which would be used time and time again over the next decades and were hugely popular during their time.

During this period, the town-tamer Western was not simply entertainment and wild tales for its own sake, but a space in which filmmakers could engage with ideological problems. These questions, as defined by scholar Richard Slotkin, were: "What is the proper balance between the rights of the individual citizen and the interests and opinions of the majority? Between the ideal of justice and the practical operation of the laws? Between the property rights of the haves and the legitimate needs of the have-nots?" (Gunfighter Nation, p. 379)

In the view of the classic town-tamer Westerns, the outlaw element represented not only social injustice but also backwardness. The answer was therefore that through violence, the hero had to defeat them and thus empower the ordinary townsfolk, who were often portrayed as cowardly or inept, to bring "progress" (or "civilization") to their town and subsequently to the American Frontier. The notion that violence against uncivilized subjects (whether that be the indigenous population of the Americas or brutal outlaws/bandits) is the key to progress lies at the very heart of the creation of the Wild West that took place in the late 19th century. I talk more about these origins here.

Over the decades, the town-tamer Western became simply one of many different sub-categories of Westerns (such as the cavalry Western, the Gunfighter Western, the Outlaw Western, etc.) that populated the stories in cinema and television. Like most historical films, they reflected contemporary anxieties, such as Tombstone (1993) which is more reflective about urban gang violence in the United States of the 1990s than the actual Earp-Clanton feud (1879-1882), and attempted to provide answers to these problems.

While this is undoubtedly a disappointing answer, the construction of the Wild West in the 19th century and the fact that we today associate supposed historical images of that period with completely fictitious scenarios (whether that be the high noon duel or the expert gunfighter) is an interesting insight into how powerful historical memory can be and how important popular culture is in spreading specific images about the past.