I also play darts (in bars)... I am pretty good. About 4-5 years ago, it became officially a sport, I was shocked.
All in all, I have had games I have sweated more playing chess than darts. I don't think that a sport has anything to do with being "physical."
By the official definition a "sport," I believe Chess qualifies. It it technically physical (you move pieces) and has a standard set of rules.
Anyway, if it is a not a "sport" yet, it will be soon enough, after all, darts is!
I guess when you are Checkmated then it's called Chess.