Bastyon has a connection to Internet, but you can install the application and turn off all your browsers – it will still work.

Turning off all your browsers? Browsers don't provide you any internet access, so why would it matter if you "turn them off"? If you're not connected to the internet, will it still "work"?

When you lose internet access, you lose access to Bastyon, BTC, and IRC because they are not on separate, parallel network infrastructure from the internet, but are only on the internet. Since you do not have access to a separate, parallel network to the internet, if they were on separate, parallel networks you would not have access to them. You access them through the internet, which they are all part of.

No, you can't access Bitcoin without access to the internet.

https://cryptomode.com/4-ways-to-use-the-bitcoin-network-without-an-internet-connection/

Every Single one of those solutions depends entirely on the internet which is what miners use to verify and process transactions, without which niet access.