I got my start in disc sports playing ultimate for my college club team after I couldn't hack it on the basketball court anymore. It's definitely a serious competitive sport at multiple levels as well as a popular recreational sport.
To the OP, I think a big part of what drove the development of ultimate in an organized way was the fact that it became so popular in universities along with the fact that it is a team sport. It was easy enough to transition this game into a more organized activity and compete with other teams/schools, so much so that at many of the tournaments I used to go to, there were multiple high school teams competing, organized by their schools, traveling and competing with school funds. At this point, it's basically one step away from becoming a NCAA sanctioned sport.
Disc golf could be there too if we continue to push for legitimacy with the younger kids, create leagues, get the parents on board that this is a legitimate activity for their children to participate in. Schools will continue to take notice and form clubs/teams/leagues and drive the sport. Ultimate just had a bit of a head start because it has the familiar elements of a team passing something to each other on a field with end zones.