I never even thought of it AS Pokemon storage until it was pointed out to me. But that needs to return from the Diamond & Pearl days. I loved being able to see all the Pokémon interact with each other, being able to interact with them, seeing them get together to do random silly things (like form a clock, standing on each other to form a tower, having a group flying session, etc.).
I think Pokemon Stadium did it better. You didn't even need to keep the same save file, and it traded fairly fast if I recall.
IIRC Pokemon Stadium only accessed your Game Boy (Color) games' boxes, and didn't have a box of its own. I think you're thinking of Pokemon Box Ruby & Sapphire, which, as you said, allowed you to withdraw a Pokemon using any game, no matter what game initially deposited said Pokemon. It was good for moving large amounts of Pokemon between games because you didn't have to watch the long trading animation.
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