Some people will probably balk at this, but I just leave my Game Genie in at all times. It will bend the pins a little more than a normal game, but I figure as long as I'm not removing it, constantly, it shouldn't matter. I modified the Game Genie so that the lid closes when I don't have a game inserted. It only takes a small amount of care to remove the game without pulling the Game Genie out. I would be willing to bet that I have a 95% success rate for starting games in the first try, maybe better. Plus, if you dont' want to use the Game Genie, you can just skip past the code screen.
To answer your question, I've heard that if you get a new pin connector that allows you to play games without pushing them down, you should play them that way. It should be less wear and tear on the pin connector. However, those aftermarket pin connectors never seem to last very long no matter how you take care of them.