I see your objection as taking the concept of individual ownership a tad too literally.
First, there's the family (or guardian) unit. Unless you're advising every individual person on Earth to run their own individual computing infrastructure.
Kragen's argument isn't for an atomized Internet, but for a decentralized one. If that means that there are local hubs or clusters that a neighborhood of 10 y.o.s could access and use, it would be a vastly better world than one in which only a handful of giants provided equivalent services.
You're also showing an impressive lack of imagination in terms of what it is to have an online presence. A decentralized structure with a cache-and-forward, high-replication, mesh-net structure could well function from minimal personal investments as a point-of-entry. A smartphone, or even feature phone, could well be sufficient hardware to enter into the system. I'm seeing Android devices priced in the $20-40 range. There are very inexpensive servers which offer similar functionality.
There's another economic class which is comparable in means to your hypothetical 10 year old as well: the vast underclasses of the world, with annual incomes ranging from $200 - $2000/year. Even among these, access to minimal levels of technology is highly sought simply because it hugely reduces the costs, and provides tremendous values, from communications.
Outernet (featured a few times on HN) is one example of an information service which is aimed at this population, with concerns over cost, energy access, censorship, surveillance, and content quality. It's a broadcast mechanism, but with some sort of mesh-net response network, it could well turn into a bidirectional system (there's also investigation of making the service itself bidirection).