Plus we shouldn't forget about the "open" part. It's hardly an open system if people are forced to transact off-chain because on-chain is constantly full and prohibitively expensive.
Wrong. Once you are actually on LN, i.e. you have locked in funds, transacting is
both faster and cheaper than doing so on-chain.
It's faster and cheaper for repeated payments to and from the same person, sure, but that's not relevant to every transaction. And when someone wants to settle on-chain to send those coins to someone else not using lightning, you'll still have to wait for a block when you close the payment channel and settle
before you can spend those coins elsewhere.
I also wouldn't mind some clarification regarding which party pays the miner fee when the balance is settled on-chain. I don't see people discussing that in great detail. The whitepaper states that:
By having knowledge of who broadcast a transaction and the ability to ascribe blame, a third party service can be used to hold fees in a 2-of-3 multisig escrow. If one wishes to broadcast the transaction chain instead of agreeing to do a Funding Close or replacement with a new Commitment Transaction, one would communicate with the third party and broadcast the chain to the blockchain.
If the fees are held in advance, how is one supposed to know what the appropriate fee will actually be by the time the channel is closed?