Maybe the pendulum could swing back the other way slightly and we could have room for an older Yahoo-style indexed search engine. This wouldn't be something that would be browsed instead of search, but if you had a general categories list indexed, say 'finance' or 'graphic design', users could enter those sections and search sites categorised accordingly. Perhaps some ranking could be done based on search terms used within a category and what site a user ends up visiting (e.g. many users end up visiting a certain stackexchange link when searching for 'parsing json in python' within the 'computer programming' category, and so its rank increases). Heck instead of ads, maybe each category page could have a list of 'popular / trending sites on this topic' section at the topic that pages could pay to be placed in.

Not sure if this solves any user problem to be honest, and the idea is only appealing to me because I think some amount of domain expertise and human curation could go into categorising pages. While this sounds (and no doubt would be) labour intensive, if we consider the number of domains that users actually visit when conducting a search (i.e. ignoring anything past page 1 of google) then perhaps it's not so extreme.