> you'd have a huge amount of money tied up in this network, and people would be spending it as if it were real bitcoin when it isn't
Note that when you have 1 LN channel open, that doesn't mean you can only transact with 1 person. You only need 1 channel open to get on the network, and then you can transact with everybody else on the network.
How is it not "real bitcoin"?
> the double spending problem is truly only solved when the channels are closed.
How are you going to double-spend on a LN channel? Your counterparty knows the state of the channel just as well as you do. You physically can't update the channel state without your counterparty's approval. Part of updating the channel state is revealing to your counterparty the secret hash that will allow them to steal the money from the previous state if you broadcast that previous state's HTLC instead of the updated state.
The double-spending problem is solved by it being possible to settle the channel on the blockchain. The blockchain is still required, but it's not like you can cheaply or easily double-spend a channel at any point in time before it's settled on the chain. If you try to do that, your counterparty will just take all of the money.