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.
LN's fee market is an interesting one. Rising demand causing individual fees to rise would actually be a bad thing and is hopefully not what we end up with. Ideally, you don't want fees to rise, but rather transaction throughput to increase by having balances flow in both directions of a channel rather than just one. This way individual fees per transaction stay low while node operator profit increases. Whether that's attainable remains to be seen. I do believe it's possible though, as there's an incentive for node operators to learn how to manage their channels most effectively.
That's not how Lightning works. LN doesn't route UTXO's, that's not how it operates. It is a network of peer-to-peer payment channels. You pay off-chain directly to a peer, but you can also pay an intermediary which then pays the destination. If they don't forward the payment, it's not like 'held in custody', instead you get a route fail error and your node tries a different intermediary until it finds one which works. Worst case, the payment fails and you get your money back (in fact, it never left your wallet until destination is reached).
I think what Wind_FURY is referring to is the funds you are left with once the anonymous node closes the channel after pushing their coins through your node. That on-chain settlement transaction will leave you, as an entry node, with the coins that the anonymous node wanted to get rid of.
