Btc and crypto wallets have offline transactions system.
So they not relied on the global general publiick internet.
This is incorrect.
Bitcoin wallets have offline "signing" capabilities. But they can not create a transaction, broadcast it to the rest of the network or sync their chain with the network without the internet.
And to maintane this independent network the very small fee will be detucted out of each transactions.
The independent network could be some boxes around the globe wich working with radio-signale completely independent of the general global internet
It is a very bad idea to create a bitcoin specific network that relies on fees it receives from bitcoin transactions to operate because it could create a centralized point of failure. Not to mention that a dedicated network could require dedicated equipment that people don't already have.
What we need is a network that already exists and is used and doesn't need any special equipment. This is why the Internet works well, we already use it and have devices to connect to the internet. So bitcoin could be built on top of it. An alternative should be similar.