The internet is the result of networks connecting to other networks. If you aim to replace that, then how? Making new networks just expands the internet, making new interconnections just makes it more meshed.

You would have to make networks not connected to the internet but interconnected with each other. That's expensive and all the economic network effects are against you. You probably won't have many users connected and not many services either.

But let's say you did it, what exactly is the benefit of a second internet? Would you be banning some networks from connecting to your mesh? What if one network in your internet connects to the normal Internet anyway? What sort of technologies and services would there be, just the normal ones, then what changes?

Honestly I don't see the point. A concentration of economic power and influence over web technologies is the issue. The internet works fine, and we make it work every day (my specific corner being research networks in Switzerland). You need to change the producer and consumer behaviour of people and companies using the internet, not the internet itself.