Darts is a sport, as much as chess is a sport. Both involve lots of training, although vastly different. I used to play darts for a team in my local area and I was surprised how seriously players took it, and that was in a Pub league.
A sport doesn’t, by definition, mean that the player is physically fit. For example, I would say that sumo wrestling is a sport and those guys aren’t exactly buff!
I think darts requires skill and strategy, so to me that makes it a sport.
Just because it does not involve running does not make it not a sport. you can say the same about snooker, pool, shooting, fencing, bowling, esports etc.
I was watching a darts competition for the world title. Athletes may not be pumped-up athletes, but their strength is in precision. It's very exciting when the leader is constantly changing.