However, the internet has shown that this does not happen.
Only because we're not doing it right. The problem with using billions of monkeys, is that you're required to keep the monkeys isolated from one another, in order to have their typing remain random. If the monkeys are able to read one another's typing, they will form patterns together. They'll learn, invent culture (i.e. spread memes), trade typing duties for sexual favors, cheat by photocopying previously-typed pages, etc. All these things remove much of the entropy needed to eventually recreate Shakespeare. They start working on problems of their own, seperate from the Recreate Shakespeare project. Or even if they remain loyal to the project's goals, the learning will guide them into local optima, when what we need them to do is continue to type randomly, all in parallel.
This is why I propose using simulated virtual monkeys, each in their own sandbox. The monkeys need to remain isolated and free of anything which might incentivize non-random typing. With simulated monkeys, we can do this!
(BTW, it would only be fair to point out that a competing research team claims they have found a faster and more efficient means of recreating the works of Shakespeare, using a key-reference system where using the title of the work, it is looked up in a memory bank and read out. This all sounds too complated to me, though, so I'm sticking with the simulated monkeys.)