Really interesting thread here...
I think that Jesus would have been a gamer if games weren't so closely tied to organized [Roman] religion. Certainly He knew how to have fun (turning water into wine isn't exactly a joyless sort of thing to do).
More importantly, if you look at "Homo Ludens", some interesting insights present themselves. Huizinga notes that organized religion tends to become a kind of game, a "magic circle" unto itself. You enter the game world, and you have awesome or not-so-awesome experiences, and then you leave, and your "real" world doesn't really change at all. Doesn't that sound suspiciously like the kind of experience a lot of people have at church?
So I think Jesus came partly to abolish one kind of magic circle, the kind which divides organized religion from everyday life. He came to tell people "hey, if you act like a saint in the temple and an asshole the rest of the week, you're still an asshole".
At the same time, He seemed to embrace other kinds of magic circles, while scratching little holes in the edges. I'm talking about parables, of course: a parable or story told "live", where you get to interact with the storyteller, is really another kind of magic circle, like a game.
In the end, I think Jesus was not a gamer in the sense that He was pretty much against the "gaming" of everyday life, the compartmentalization of work and family and religion into their own little boxes. On the other hand, He wasn't averse to the use of fiction and fantasy, of putting his listeners in the shoes of someone else. In other words... Jesus may or may not have been a gamer, but he was almost certainly a DM.