That web still exists: look at Wikipedia, one of the largest websites, run entirely not-for-profit with the aim of sharing the sum of human knowledge. Nothing stops people from starting not-for-profit websites today and only linking to other not-for-profit websites. Running a website has never been cheaper and search engines are light years ahead of anything that existed then.
They may never be as popular as Facebook and Twitter, but that's because the web has grown in audience to people looking for something similar to watching TV, reading newspapers and going to the pub. Those activities will always be more popular than people hacking in a garage, but they can coexist.
