I address his question in my Virginia City: Secrets of a Western Past (2012). In northern climates, batwing doors would never have worked; saloons in the Intermountain West generally had full-length doors. This is largely a product of movies and television. In the warm Southwest, one might have encountered batwing doors, but they would always have been reinforced by full, locking doors for when the saloon was closed (or when wind and dust was an issue or during cold winter evenings).

Another thing that Hollywood gives us is a different orientation: doors open in films on the longest wall. Saloons, in fact, were orientated to have the least front footage since that was the expensive real estate, so doors opened to long narrow saloons with the saloon leading back to the back wall. But that doesn't work cinemagaphically, so Hollywood turned the saloon so the doors opened to the broad bar on the opposite wall.