It's worth remembering that there is an important difference between "the Web" and "the Internet". This is the only way to understand something like when Alan Kay says, "The Internet was done so well that most people think of it as a natural resource like the Pacific Ocean, rather than something that was man-made. When was the last time a technology with a scale like that was so error-free? The Web, in comparison, is a joke. The Web was done by amateurs."
If you're less clear on what this distinction is: HTML+CSS+JS transmitted over HTTP based on domains looked up via DNS -- this is the Web. Things like email and BitTorrent and peer-to-peer networks and older things like Usenet newsgroups have always existed on the Internet parallel to the Web, to say nothing of, say, SSH and ping and other developer-friendly utilities, or newcomers like BitCoin.
I've given a chunk of thought to how we might reboot the Web as a new service living on the Internet and I've come away with more questions than answers. What the Web has done extremely well is to convince a bunch of companies that they need to have a Web Server living on the thing and hosting their content. What it has done extremely poorly are these problems which maybe we don't even think about: link-rot, the fact that you can't throw a dime at a YouTube video you liked and trust that the content creator will get 8 cents and YouTube will get 2, the fact that nobody can directly advertise to you, "watch this video and we'll throw a dime at you!", the fact that you can't actually just fire up your browser and start publishing a feed of videos and articles that you like interspersed with a bunch of content you're writing -- you instead need to sign up with some social network or blogging platform and then play by their rules. The problem gets even harder when you think about systematically purging illegal content and discouraging spam and so forth -- indeed the key trait is to be able to say "I want to identify who is holding onto this content and go after them via non-technological enforcement measures" and such. One gets the impression that the Web is a very niche tradeoff in a vast configuration space and that there are many other possibilities -- but it's hard to see one that does something so fundamentally different to the Web that people will instantly be addicted, "THIS is what I want."