The deeper problem is advertising. It is sort of a prisoner's dilemma: all commercial entities have a shouting contest to attract customer attention. It's expensive for everybody. If we could kill advertisement permanently, we can have an internet as described in the question. This will almost be like an emergent feature of the internet.
Other than a completely new approach for producing value such as the 'Freeism' one described in the article suggested in this comment https://news.ycombinator.com/item?id=20282851 (which I hadn't time to read yet and hence I'm neither in favour of or against) this simply boils down to the questions of who will pay for relevant content and what the business model will be. By and large, people don't seem to be willing…
I don't think the causality is right here. People might not be willing to pay for content on the web because advertising enables competitors to offer content for free. If you removed that option, if people had no choice but to pay, it might just turn out that people would pay.