I don’t know about running the whole internet over peer-to-peer network, but my home server is pretty much the ‘main’ computer and while phones an laptops obviously have data locally it’s also synced to the server so losing one mobile device isn’t really a big deal (besides money to get a new one). Immich for photos, nextcloud for other data, radicale for contacts and calendar and self hosted imap-server for emails.
Obviously the devices are still very much personal, but it’s easy enough to wipe and start over if needed. For remote wipe I still need to rely with google on phone and with laptop there’s currently no way to remote wipe it but it’s running with encrypted drive anyway so it’s only the monetary value of the thing in case it’s lost.
There are various mesh-network projects around and it’s better than nothing, but their issues tend to be pretty low bandwidth and physically limited area. Wifi-mesh in a somewhat densely populated area is technically possible, but technology says that you need to be pretty close (100m give or take) to the next node. On rural areas people have built pretty long range wireless jumps without ISPs but hardware requirements for those are a bit different and you’re relying heavily on the node next to you in upstream direction.
Then there’s things like LoRa Networking, but their bandwidth is very small and it’s really only suitable for SMS-style messaging with pretty low traffic, but it can reach up to 10km between nodes. AX.25 over amateur radio has range up to hundreds of kilometers, but it’s also pretty slow (~1kbps).
So, in practise, the best would be to use something like NNTP and distributed servers across the mesh network where you’re less dependent on long range high speed communications. Modern web experience or instant messaging just isn’t really feasible over any mesh network with current consumer-grade hardware.