I think that there's room for us to abstract away branded search under a new category, but I don't think new browsers are the answer.
People rely on brands when they have a product whose quality is important to them, but is difficult to discern from casual inspection. So for example, the strongest brands tend to be in markets like soda, foods, and cigarettes, because you're putting this stuff in your body and yet you can't tell what went into it. Branded medicine can fetch prices several times higher than generics, despite being exactly the same stuff. Things like software - Microsoft and Google - have middle-ground brand power, where quality is important (but not all-important) and you have few indications for overall software quality at purchase time. Things like airlines have virtually no branding power, because the trip is over in 5 hours, you've gotten to your destination, and you quickly forget about just how unpleasant the flight over was.
Following this, there are two ways to cut brand power of entrenched competitor:
1.) Make consumers care less about product quality. Cunard and White Star were done in by the airline industry, because by cutting trip duration from 5 days to 5 hours, they made consumers ambivalent about trip quality.
2.) Give consumers better information about the actual quality of the product. Google destroyed branded directory sites like Yahoo because they showed everyone just how many websites the directories were missing, and let them evaluate for themselves the quality of the sites.
To achieve #2, you could build some software - either a browser plugin or a meta-search engine - that "rates" each result on an unbranded results page, saying "This is filled with adsense" or "this information is false" or "this was copied from here". If you could convince consumers to trust the ratings more than the original search, you'd break the brand power of the original search engine. However, that's probably at least as hard a problem, if not harder, than building the search engine itself.
FaceBook and other social networks become dominant because of network effects, not because of brands. They don't let you export your friends, so you become trapped there, without much value anywhere else. FriendFeed tried to break that cycle with their meta-social-network, but it didn't work, probably because they didn't add any value besides what the social network itself had. In cases where new software really is superior, I think it's possible to convince whole cliques to move over at once, though, and that can break the back of the entrenched competitor. I saw it happen with LiveJournal, and then again with MySpace.