LN does provide certain privacy benefits, but the culture of the spec attempts to make it a global network with complex routing and thus some privacy is lost. Generally, direct P2P is an optimal privacy model, but once you start routing and sharing route data and listening for requests etc, things get tricky.

LN could and will be better, Synonym actually has some ideas about ways to improve it that we will probably float to the community this year.

Mixing is mostly dumb, I believe privacy is lost not gained, and that creating extra metadata is bad. That said, LN route mixing is interesting, and even more so once we have tokens and multiple assets to swap ;)