A search engine that functioned similar to old web rings of the 90s, where each website effectively endorses a collection of other websites and the search starts with a list of user-approved trunk sites and gives results that branch out from those sites, could be an interesting version of this. Would require some initial work on the user's part (flagging sites they trusted and liked to use as the trunk of the tree) and on websites' part (each including a list of other sites that they endorse), but it could be a much nicer way to browse. Obviously search results would be very limited, it wouldn't be good for looking up specific information on new topics, but if you had, for example, a tree of sites with gaming information and news, and you were looking for articles about a new console release, presumably it could help you find it. Think of it like current web search but which will only provide results from sites that exist in your tree.

Show how many steps each result is removed from the trunk, show which site each referral came from, and give the option to blacklist sites that you don't want to see (and in turn, all of their referrals), and over time you could end up with something approaching the old internet, at least in so far as you saw.