Mesh network can work but they are like connecting one node to another and without internet the range will be too low for example the Bluetooth can be used but the speed of BT is not fast enough to synchronize the blockchain data from every node quickly, the data won't be moving from one node to another and it won't update the whole network and this will just delay everything.

On top of that, with a mesh network with BT, the range is a big issue, you can connect nodes in one place, tell me how many people are running nodes around you? Don't know, but if in 100 KM range no one is running a node, then you won't be able to broadcast any transaction and make any so you are out.

Therefore internet with speed and range is necessary, we can use wires, but that is going to take time but if we have to use internet cables, then why not use the internet itself because the internet I am using and you are using connected me to the whole world with wires.