Where would that put Sam Peckinpah's best Westerns (Ride the Hight Country, The Wild Bunch, Ballad of Cable Hogue), where the automobile is an important symbol of the passing of the old West? And hey, what about Roy Rogers and his trusty jeep? :-)
Well, for that matter, where would that put "Bad News Bears", "Every Which Way But Loose" and "Raising Arizona"? They were all filmed in the west.
What makes a western a "western" to you? Cowboy hats and clothes? Guns? Drinking in a rustic bar?
"Monte Walsh" or "Son of Paleface" is about where I would cut the western umbilical. Where the auto is just being introduced into the west. When only a few people are riding horses, it isn't a western. Your mileage may vary.
BTW: I loved the old Roy Rodgers/Gene Autry serials as a kid but really can't watch them now for other than nostalgia reasons (pro USA, good guys are the best)