I don't know what famous people you're talking about ... I'm assuming we're leaving out "celebrities" and concentrating on "useful" famous people in business/academia? If so, you'd be surprised at how much correspondence they can have.
Not to mention I'm not sure if you realize how many of the "emails" on things like VC pages are bogus on purpose. Like when they say, "mail your business plans attention "Business Plan Manager" ... that's usually code for "just toss this in the paper shredder.
Here: test this out. Think of 5 famous people and email them something you think surely would catch their attention. See if you can find their real emails and if they bother reading it. Use one of those "verification of opened email" services.
It just doesn't happen that often. If anything it gets sent to an assistant who sends it to another assistant who sends it to the intern who gives it to his little sister who gives it to the cat.
I think this would be a good "idea" if not necessarily one that translates into a business.