[aleph_minus_one]

I have the impression that the author does not want to build a parallel internet (which involves whole new protocols replacing, say, IPv4, IPv6, TCP, UDP, ICMPv4, ICMPv6, DNS, perhaps TLS, ...), but rather another application-level protocol replacing HTTP and the web.

Remember that the world wide web is just one of insanely many application-level protocols that can be run over the internet infrastructure.

[initramfs]

Correct. The idea is to narrow the application access view from the desktop or home screen of a mobile device to a limited set of optimized apps and protocols that can traverse the web. It's kind of like a dedicated bike lane alongside a boulevard in a major city. One can bike as fast if not faster than cars, but they'd need to adhere to the rules of the bike path.