I agree 1000% that people compartmentalize based on functionality, which is why Google shouldn't be afraid of Facebook on search (that and Facebook not having search and search being difficult).
Could I create a reasonable search engine only using URLs people have "liked" and weight search results based on number of likes and whether people in my friends list have liked it?
There already is a very good search on Facebook and the fact that it is powered by Bing is unknown and irrelevant to 95% of the people. However it hasn't made a dent in search, so you have to seriously ask why that is. If you believe in Facebook as a search engine (I don't), the only real answer is because it's not powered by Likes. But to believe that, you have to believe that people have a major affinity for Likes, enough to throw away their old search habits in favor of it.