I don't think you should dismiss genres based on preconcieved notions (Morari). I never played MMOs (until WoW) but I always at least tried them. I played UO, EQ, and FFXI for a few sessions before I decided I didn't like them. WoW actually got the formula right, IMO. You can learn a lot of interesting things that you can apply to how you play other games if you spend the time exploring and thinking about how things work, particularly in the realms of how statistics work in gameplay, and how monsters choose targets, working cooperatively with other people, etc - after I had played WoW for a while, I was able to pick up on a few certain gameplay elements in other games much faster. Examples: immediately being able to imagine a bell curve of how much time it would take me to get some dumb random sword drop in Castlevania, or working with a friend in a co-op game to "kite" a boss enemy in Random Co-op Adventure Game.
Of course, the same is true in a lot of other genres once you start to play above a certain level. Fighting games will make you think about things like anticipating moves, thinking about move "priority", and considering stuff like frame data/move recovery. Seriously, you can apply this stuff to non-fighting games, like, say, versus mode in Bomberman 64. If you'd have happened to have played a lot of Tekken, you would pick up on little things like the number of frames it takes you to kick a bomb or the exact margin of invincibility you have after you lose a heart. That stuff wins matches.
One more example: if you learn about racing lines from a game like Gran Turismo (which are a lot more relevant to playing well than in Mario Kart or F-Zero), you can apply it to getting from point A to point B in virtually any other game. For example, cutting corners on the path in STV to chase down a lowbie player in WoW so that you can gank them and spit on their corpse. Sure, racing lines are "real world" knowledge, but by practicing it in a video game for hours and hours, you can recognize opportunities and implement it faster/more effectively than any old douchebag that doesn't play a lot of different games.
Playing a lot of games will help you identify the nuts and bolts that the gameplay is made of. I equate hardcore with skilled, and I'm quite certain that playing a wide variety of games will make you better at video games in general if you're actively thinking about what you're doing and you know how to apply that knowledge.
I don't play EA Sports games, but I got Tiger Woods for cheap and I actually enjoyed it a bit.