Interesting.
Is ballet... sport? Chinese dance? Circus acrobatics?
It kind of depends on what your definition is. If competition is a part of the definition, than professional wrestling, as a staged, scripted activity, doesn't fit that definition.
If your definition is simply highly athletic performance/entertainment activity, then professional wrestling qualifies, as do the things I noted above. On the other hand, poker tournaments, esports, etc., then do not qualify.
My suspicion is that, to most observers, the definition is slippery and moving - "sport" is in the eye of the beholder.