the network is centralized. To transfer BTC, you just have to send your Bitcoin to the private group address, and specify your other address on the layer 2.
It doesn't need to be on layer 2 or there is no need for mining.
This can happen easily on bitcoin blockchain without needing mining. Similar to how Tether works on bitcoin network. It is centralized and you only transfer Tether tokens on bitcoin when your transaction contains a special script in its outputs that the Omni layer can only evaluate.
You can create 21 million Belisarius Token out of thin air and promise that it will be converted to bitcoin 1:1. Then people can send their bitcoins to a specific address to "lock" them and receive Belisarius Token.
On second layer this could be repeated without needing to send an on-chain transaction each time. The user sends their coin to a specific address (basically locks them in a special script/smart contract) and gets to spend the same amount on that second layer.
There is no mining on second layer, just nodes that have access to bitcoin blockchain and can verify if the amount user is spending is locked in that smart contract or not and be able to create many transactions among themselves and only settle it on-chain once.
Lightning would have been good, but I have one big problem: the value. 1 BTC ≠ 1 LBTC. Because of this, it is possible to lose a lot of money due to price fluctuations.
Any amount of bitcoin on Lightning Network is equal to any amount of bitcoin on main chain! There is no difference.
I think you are confusing LN with shittokens on altcoins that call themselves "XBTC" such as the WBTC on Ethereum platform. Those could fluctuate in value or disappear but LN can not.