I understand this point of view but I think there's a difference in the way media is enjoyed between movies/music and games in that movies and music are far more universal.
Simply "liking" movies and music really doesn't need a label because nearly everyone likes movies and music. I've heard anecdotally of people who don't like one or the other, but it's like hearing about your cousin's girlfriend's ex roommate from college who totally loved peanut butter and dried cod sandwhiches with pickle relish. I've never met that person or anyone with that kind of experience and doubt the veracity of the claim.
Games, on the other hand, are different. You can't throw a rock without hitting someone who doesn't play games... Many, ponderously, proudly so. It's very polarizing in some respects. Given that, I think "gamer" still works as a term for "someone who plays games".
The caveat there is I think it definitely has a shelf life. As games become more and more commonplace, it's certainly on its way to losing any sort of meaning. I don't think we're there yet, but I understand why others might.
As a corollary, I think we've already reached that point with "gamer culture" (or more appropriately "gamer subculture"). As the generation who grew up with video games which defined "gamer culture" in the past now has children of their own, "gamers" now include vastly different generations with a much greater spectrum of language, norms, symbols, etc. to the point where there is no single "gamer culture". Some will say there never was, but that's probably a debate best left for another conversation.