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/
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.

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Lightning Privacy 101: The basics of second-layer bitcoin transactions
Bitcoin's Lightning Network is a major step forward in improving the privacy of bitcoin transactions. The Lightning Network has several features th...