No. It means that it COULD take longer than the age of the universe (and even that it’s expected to be so), but if you’re “randomly” choosing items from a set, this means that choosing any item on such set has the same probability as any other, so it also could take you (though with much less probability) less than 3 attempts until you find one that isn’t empty.

Maybe the bitcoin protocol actually guarantees that the drawer is in fact empty, but that’s not what you wrote.