But Lightning nodes charge fees if transactions are routed through them right? Then if there are high enough entities who are holding UTXOs they don’t like, demand will rise and therefore fees will rise. Unwanted UTXOs can merely be routed through Lightning, and leave those unwanted UTXOs with the routing node.
Observers would call this mixing, I believe it is increasing the anon-set.
Yes, Lightning Network is likely going to increase the anon-set and thus help with Bitcoin's fungibility. I'm not sure if it would be a good thing though if LN became just one big decentralized tumbler. Not that I really see this happening.
If it would incentivize Lightning node operators to increase liquidity in their channels, then I believe it’s going to be a net positive.
Lightning simply for payments is currently not happening, everyone prefers to use onchain. But it will happen for something else, another form of utility for users.