Game data is definitely stored on the game carts themselves, not the system.
Turning on the system once for a few seconds to see if it has any dead pixels isn't going to make your battery never work again or anything, just don't turn it on and immediately play Phoenix Wright for 2 hours until the battery dies without ever plugging it in like Chris's friend did with Advance Wars the day he bought his DS. I mean, surely the systems are turned on once at the end of the assembly line to make sure they are worth putting in a box, etc etc. You should be fine.
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