I don't know about the Wild West specifically but this was far from an unheard of thing in 1880s America. If "running for pleasure" was weird it would be because they already had a similar sport: competitive walking (essentially very long distance travel around a track was america's first spectator sport). It's possible it would be alien because competitive pedestrian(ing?) was mainly a city sport. So this seems anachronistic to me.

Also as the film Buch Cassidy memorably shows this was also the era the bicycle started to become mainstream. http://www.britannica.com/EBchecked/topic/493834/recreation

http://www.npr.org/2014/04/03/297327865/in-the-1870s-and-80s-being-a-pedestrian-was-anything-but

also discussed on backstory radio

http://backstoryradio.org/shows/u-s-vs-them-2/

It includes an interview with Matthew Algeo who wrote the book (literally) on the walking craze

so i can't exactly refute this point; however, given the circumstantial evidence it seems unlikely by 1885 this would be a weird idea.