No it doesn't. Especially when hinging on that caveat "if used correctly" which most users won't.
https://voltage.cloud/blog/lightning-network-faq/lightning-network-privacy-explainer/
https://blog.keys.casa/lightning-privacy-101/
Monero is just another trade off, like lightning/liquid, which you have no problem using. Except, unlike both, it doesn't compromise most major value props of Bitcoin. "I don't recommend keeping a lot of funds there" - Ok, just apply this same advice to Monero. Done.