I can't imagine how this is possible. Imagine I have a string of words (a quote from a book or an article, a fragment of an error message, etc), and I want to find the full text where it appears (or pages discussing it). How would you do that without a search engine?
I think OP's idea is that search services would be built into the Internet, and not provided by a third party. That is, when a website is published or updated, it is somehow instantly indexed and made available for search as a feature of the platform on which it was published.
Not a chance it would survive. Google has enough problems fighting SEO right now and they don't publish their algorithm and have incredibly deep pockets.
Personally, I don't want "The page rank algorithm" I want 100 page rank algorithms made by 100 people. Transparency is important, but I think competition is more important.
You ask a question (possibly on Stack-Something) and don't get ridiculed for not using Google, since you live in a world where search engines don't exist.
And how would you find out if that question has been answered before? That would only work if there was single unified centralised question site. And then we are pretty much back at Google's single search field.
We could have website-centered search engines. You ask the question on whatsthatquote.com and find out if someone has already asked it. If yes, you have your answer, if not someone answers and no one is annoyed.
Stack overflow does that. You don't get ridiculed for asking a quesiton on so that has already been asked on another website you don't know of.
I guess that would be the age of smaller communities centerd around a few websites only? Maybe, I don't know if we can consider google as enabling a real global community as of today. I pretty much browse around the same websites. Anything I want to find without a precise source of information in mind, I use google and stumble upon ads and ads and sometimes ads, but rarely an answer.
I sometimes still search stuff manually browsing through websites indexes. Some things are difficult to find with keywords. Equations of which the name you forgot. Movies with a plot so generic billions of result would be associated with it on a search engine. That piece of music of which you could write the notes on a sheet but don't remember the title.
The most informative answers I've encountered on StackOverflow are either a product of research (benchmarking, analyzing multiple sources) or very specific knowledge, sometimes written by the author of the framework/library in question. I'm not sure your analogy applies since these answers demand substantially more effort than the (usually) predictable and repetitive questions teachers face in class.
At some point someone has to write it, like school textbooks. After that your just looking at distributing that knowledge. The replacement for search engines solves the latter problem.