Nothing is truly private. Lightning is fairly private, and cashu is even more private, but in order for them to be at all trustworthy, they must build in layers atop one another.
Discussion
If you have anonymity.. you really don't need privacy .. in fact privacy invites attention !
No, anonymity in public attracts attention. Privacy means you or your actions are not in the public eye.
Taking actions in public that preserve your anonymity: suspicious.
Taking actions in public that preserve your privacy: oxymoron?