YouTube wasn't considered crazy at all by the time Google bought it. It was clear by then that it was a runaway success, so clear that even traditional media companies like Viacom and News Corp were bidding for it. It certainly wasn't some small company that Google grew from nothing into dominance; it was a billion-dollar company that had already won its space.
Some articles from the time:
http://money.cnn.com/2006/10/09/technology/google_youtube/
http://techcrunch.com/2006/10/09/google-has-acquired-youtube...