Full privacy is an incredibly high bar. I would say you need at least a mixnet style design, in which every participant does exactly the same thing at exactly the same time to the outside world (and that 'thing' is encrypted, and if you don't need to do that 'thing' (send a message), you still pretend to do it) to start claiming anything like 'my anonymity set is the entire set of users of the protocol'.

Even something like zcash which has a vastly more advanced privacy tech setup than Bitcoin, still suffers from not following that strict model; there, metadata can leak from the number of transaction inputs, even though the tx graph is 100% obscured.

As for lightning, it has great (but not perfect) sender anonymity but today, fairly weak receiver anonymity, but even that 'weak' is better than on-chain (at least, using Tor at network layer as most users do) because there isn't a publically readable, permanently stored, tx graph.