Other way around, I think. Competitive video games ought to be called "competitive games," like chess. The whole "e-sport" label is stupid: "sport" has always meant a primarily physical competition, and competitive games of the sitting-down type--chess, checkers, Go, Scrabble, etc.--have been their own thing, with leagues and tournaments and titles, for decades or centuries before video games came around. Video games are more dependent on quick thinking and reflexes, but they're still primarily mental competition--see also speed chess, etc.
But competitive video gamers are too full of themselves to be happy rubbing shoulders with something as unhip as chess. They insist that they deserve the same (arguably equally undeserved) acclaim and money and fandom as sports, so they keep pushing the "e-sports" label and all the associated silly trappings and theatrics. After two decades of fruitless flailing, it seems like it might finally be about to stick, more's the pity.
Think of it this way: imagine if Kasparov and Fisher live streamed all of their chess practice, informal competition, and formal competition. Odds are chess fans would line up to watch.
It's just like that, but with strategic online games instead of strategic board games.
Also, some of the streamers are practically comedians, so they sometimes watch for general comedy and enjoyment as well. Or a mix.