I don't recall ever seeing in the fine print in public timetables anything about unaccompanied children. However, since the film was certainly produced with the cooperation (and possibly financial assistance?) of the Santa Fe, it can't have been in violation of established rules and practice. Note that at 3:05 the father entrusts the boy to the care of the Pullman conductor, who, unlike the train conductor, remains with the train throughout the entire journey; I believe that was fairly usual. I'm not sure how it would have worked if the boy had been traveling in coach. Some trains of that period, I think including the Chief, had a female attendant in the coaches, called a hostess or stewardess, who assisted in the care of accompanied small children; they also stayed with the train the whole way, and maybe something like that could have been done.
I must say I have trouble imagining a 10-year-old doing that totally alone; however I recall riding local trains in North Jersey when I was about that age with nothing said as long as I had a ticket.