KYC and lightning

I keep hearing that because lightning is off chain, transactions are anonymous vs the L1 blockchain which is pseudonymous. If that's true, is there a way then to cleanse/sanitize KYC Bitcoin using lightning? Ie, convert KYC Bitcoin into what is effectively non-KYC Bitcoin by running it back and forth through lightning transactions and then back to the main L1 chain for permanent storage? If coinjoin will be problematic for some exchanges (I understand there are exchanges that won't handle Bitcoin with a CJ history and they look back at least 5xns with chain analysis), then is there a way to achieve the same sanitation and removal of KYC through lightning?