LN can be more private than monero, or zcash. The use case simply hasn't been supported well yet by existing wallets and deployments. LN uses one shot onion messages which are inherently private, and using hidden services, on both payer and payee. Partly the protocol still needs more work in this direction.

Payjoin batching also can improve privacy by breaking chain analysis assumptions. And then there's bulk channel opening via coinjoins... In times like now, atomic swap into monero via lightning for paying privately makes sense, but wouldn't, if the LN privacy was sorted out properly for such a use case.