The potential issue with Rugby is whether or not it is seen as "almost" as violent as football. Certainly on the East Coast lacrosse could replace football as a fall sport. Given Roosevelt's "American" attitude I could see him promoting the "American" sport of lacrosse.
HS & College:
Fall: Lacrosse, Rugby, Soccer
Winter: Basketball, Hockey (limited), Volleyball (limited)
Spring: Baseball, Track & Field
Pro:
Fall: same as above soccer less likely
Winter: Basketball, Hockey
Spring/Summer: Baseball
The thing about Rugby is that it was considerbly less physical than football, due to the fact that it is a more free flowing game and in the early 1900's it was even less physical than it is today, there was even soccer style kicking among the forwards in the scrum and mauls back then. This was noted by many college administrators at the time, especially on the west coast. There was also the attraction of bringing in touring rugby teams from Oxford and Cambridge every few years which could add to the prestige of non Ivy league schools.
I would be interested to see the develpoment of lacrosse. Is it a strictly amateur sport like Rugby? Could a proffesional league develop or would it remain an elitist sport?
I envisage Rugby and Lacrosse as remaining extremely popular at college and high school level, getting the same crowds as college football but remaining strictly Amateur and not forming pro leagues until much later. Outside of college and high school there's the odd international touring team and maybe a California vs everyone else game.For Lacrosse an annual series against Canada will fill the gap.
Without pro foorball baseball remains the main pro sport( maybe more potential for minor leagues to grow). However I feel like there is still a gap to be filled by a pro sport in Autumn. Soccer can take some of that but maybe other sports also get in?
Basketball emerges earlier.
Another trend which died quickly but effected the rest of the world was resistance to proffessionalism in sport. Perhaps the amateur trend will remain stronger in the US for much longer