Blockchain should be transparent and be able to register all transactions. If LN allows off-chain small transactions for performance purpose, fine, but how to ensure transparency ?
I disagree with this. Bitcoin should be private and fungible, and Lightning offers significant privacy improvements. It is a plus.
To add to that: Anyone who wants to continue sending transparent Bitcoin transactions can do so by staying on-chain. Anyone who wants to remain private would currently either use a tumbler or not use Bitcoin to begin with. In other words, neither transparency nor privacy can be forced by an outside party, regardless of LN's existence.
Either way, I'd also argue that cryptocurrencies should stay private, where possible. The way I see it the main reason for transaction transparency is to ensure that the money supply is not manipulated. If we can ensure the integrity of Bitcoin's money supply without depending on transparency I see little reason not to do so.