I can't put my finger on the book right now, but IIRC there are several photos of cowboys up Montana-way (1880's - 1910 or so), wearing a sixgun in a crossdraw holster.
Yes, but the average picture did not accurately reflect the reality of the west. For example, every dude that came west bought a stylish, loud "Real Cowboy" outfit complete with guns and had his picture taken. Most photos of working cowboys at corrals, riding herd, etc., show fewer firearms. Rollins notes this fact in his study:
The Cowboy: An Unconventional History of Civilization on the Old-Time Cattle Range
Rollins participated in drives and was an educated man who went back to the range several times to study and carefully document actual cowboys. He also noted that almost all real cowboys carried full length pistols and says they knew how to shoot. He says that cowboys never carried unless they would be riding in territory expected to be dangerous or knew there were in fact dangerous animals about. He says that they did carry in town because they were showing off and that they always carried when courting because they felt it made them more attractive to the fairer sex. But when pushing cattle, he says firearms stayed in the wagons unless need dictated otherwise. So, our perception as shaped by photos of actual cowboys must take into account how many of those photos were staged. If a character wanted to get photos of real cowboys, they obliged by putting on all their crap, including guns. The amount of the book dedicated to guns is small, indicative as to their actual, not perceived, importance in the West.
Now, if you lived out there--say prospecting or living alone or such--you might be closer to a weapon at all times as you relied upon yourself more. I have known old timers and they always owned guns and could get to them, but they did not think about guns much or worry about them like we do today, it was simply a tool you could fetch and use like any other. Again, Lawmen and badmen and idiots who dressed like what they thought cowboys should be were the outliers, not the norm.