Gender: Male Location: Following the Source of Light.
True, but with 29th century technology, I don't think processing power or memory space would be a problem. They wouldn't be different personalities; they'd all be tied to one, central EMH. I bet the crew would object. One sardonic EMH is plenty, after all!
Now that you mention it, a script explanation is more likely. One like: the writers never thought about it, or the budget constrained a scene with a hundred of holodocs. I still think it would be fun.