I don't think that Yoga is the way to go. Yoga lives from doing certain things because they are traditions instead of having good reasons for doing them.
Feldenkrais would be an example of more modern body work that actually has a robust theory base behind itself.
Prasara yoga seems to provide advantages over completely traditional yoga but be phrased in way that's still target at New Agey people. Feldenkrais having been created by a physicist should be more accessible to people who see themselves as rationalists.
I lately discovered Contact Improvisation. While a lot of the crowd that dances Contact Improvisation has a New Agey background, you don't need to accept any spiritual ideas to follow the framework. The rules aren't complex. I was able to have a good dance the first time I went dancing Contact Improvisation without taking a course. That's partly because I do have other dance experience but there are other dances that I couldn't dance intuitively.
For people without dancing background I would recommend to take a course in Contact Improvisation before dancing it freely.
Scott Sonnon, the man in the prasara yoga video, takes a rationalist approach to exercise.
Here's Shiva Nata-- a sort of yoga that involves combinatoric movements. It would be easier to structure as competition-- the scoring could be based on the complexity of movement that a person could do accurately.
This post is a bit of an experiment; Most of the time, Discussion post lay out an idea and this idea then get commented upon. This post, on the other hand, will be purely about discussion on a topic. If this works out well, I'll might post more of these in the future.
On to the meat of this post:
I got this idea from a reddit post on /r/LessWrong.
To quote:
So have at it.
I only ask for one thing and that is to hold off on proposing solutions for 24 hours before giving suggestions for actual sports. In the first 24 hours, please discuss what makes current sports unappealing to rationalists and what aspects a sports designed for aspiring rationalists should have.
Edit: The 24-hour window has closed and solutions and suggestions can now be given.