Yeah I'm kind of having the same problem which made me really interested in the topic, Sketch dance was actually on my read list so knowing they have one is nice.

There is technically Megalobox if we count anime, also JoJo part 7 kind of race which offered complete freedom in vehicle selection and allows for weapons.

I have the feeling that some isekai I read had a sport that demons used to play but I can't pinpoint what it was.

I also remember a doujin that treated sex as a competition as akin to artistic gymnastics

I did try to think of competitions like the death games in Alice in Borderlands, also omniscient readers viewpoint and tower of god had many interedting competitions, but while sports are part of games, games are not all sports, then again, chess is by some considered a sport so I'm currently really confused if any of these games would actually be a sport and where the line lies. Maybe my question should be games in general...

But at the end of the day there just aren't that many games I can remember that I would intuitively consider a sport like it happens for air gear and megalobox. Its kind of disappointing since it is a pretty interesting area to explore for world building and it rarely takes much attention.