Re: New "internet"
In order to do that effectively it would require everyone to utilise the same VPN cloud (presumably from a trusted source) who is set up to assign bespoke DNS and private IP allocations (fixed or otherwise).
It can be done, but it would have to start slowly since initially there would be shag all content on it.
I would also recommend that all vpn implementations disable split-tunneling and the VPN cloud have no access to the rest of the internet except potentially via an anonymous forward proxy which also stripped out all unsafe protocols, only allow outbound connections etc. - although I would err on the side of no access at all myself - much safer.
That way, all subscribers to 'FreedomNet' could publish what they like via their own IP and register DNS entries that could even overlap internet ones without problem.
The main issue you have then is how to protect this FreedomNet from it's own subscribers - how would you control who gained access and determine if they have nefarious purposes - be they undercover agents or criminals etc.
It would cost a lot of money and involve a lot of risk and headaches for whomever sets it up - that's probably why it hasn't taken off yet*.
*I'm sure there are already networks out there like this, even ones you haven't heard of.
PS If you want to create a secure network just for trusted parties that you already know - then you could always just set up a VPN between your ADSL routers, or for the super-secure, between PC's (after applying all the OS and app hardening guides you can put your hands on).