I'm not a "gamer," but I've played my share of online games and video games, starting with an online text-based role playing game back in the very early 90s. I'm talking pre-AOL. But because of that game, I can type about 90 wpm, which served me very well in college and graduate school and then throughout my career.
I've not played any video games in several years, but my favorites were Star Wars Knights of the Old Republic (omg so fun!), Final Fantasy, and Kingdom Hearts.
Then there are the kinesthetically--involved games--the Wii fit system and games --- bowl, hula hoop, baseball, yoga, meditation, tennis, dancing, ski, and on and on. Lots of very active games.
As far as more recent games, there are so many genres, one statement cannot encompass all games. There's strategy games, puzzles of every imaginable kind...my little grand-nephew plays one where he's learning to better articulate by watching children speak on the screen while another window shows him on the camera. Very educational.
Not to mention the community aspect involved now with having to create guilds and such to team up. That involves strategy and cooperation, negotiation, leadership skills and the ability to take direction, etc.
I'm not going to keep going. There are things to learn from all games, and various levels of different kinds of skills. Of course there are crappy games out there! That's for parents to decide for their children and adults to decide for themselves.