Some Bitcoin users need anonymity, and some don't. That's why mixer companies make profits. In a full Bitcoin world nobody buying stuff like a coffee or a McDonald's would care about anonymity, like when they pay for those things with debit cards today. For transactions where people care about anonymity there's already mixers, and there would still be mixers in a full Bitcoin world.
to add to your point. in a "full bitcoin world" where LN is used for small/frequent purchases.. LN is basically a multisig address that many people put funds into and they all sign the transaction when they outputs are all going to where they should.
this is the same as mixing/coinjoin principle. thus LN is automated mixing