That's the current mythology.
"Of all the colourful horsemen in history, perhaps cowboys exert the most enduring fascination. People seem to need these archetypal heroes, even if many of them were drunkards, rapists and murderers. The first cowboys arrived in the Americas with Columbus. They were wild cattle hunters and were truly the lone frontiersmen of myth. Over the centuries, though, their freedom to roam the range and live like Indians between earth and sky has gradually been eroded, so that the modern cowboy is simply a working horseman.... "
In all fairness, it should be said that although it's a serious well researched book, by a cowboy enthusiast, the author is an English country and western singer, whose real name is Dr Samuel Hutt. His day job is gynaecology.