I can't quite follow your claim of LN being kind of a mixer where you can gain or maintain anonymity. When you open a channel you have an on-chain funding transaction with a publicly visible receiving address that with non-Schnorr signature looks a bit different than normal Bitcoin transactions (the script is different and recognizable to my knowledge).
Now when you close that channel you receive your part of the channel's balance via this funding address. As far as I understand it all addresses are somewhat linked together (UTXOs used to fund the channel, channel funding address, channel close via funding address to an UTXO in your wallet). I don't see any magic involved to link all those addresses together. More channels just add more linked addresses per channel.
The blockchain will see your funding transaction with an address and the funds it puts into the channel and the closing transaction with the final balance awarded to that same address (usually).
However, it doesn't see to whom you sent how much and when, and from whom you received how much and when.
UTXOs used to fund the channel, channel funding address, channel close via funding address to an UTXO in your wallet
It's correct that if you try opening a channel and funding it from multiple UTXOs, they are going to be linked just like in any other transaction, so that should be avoided if they're in different disjunct anonymity sets.
More channels just add more linked addresses per channel.
This is a wrong understanding. Each channel is independent from another and they are not linked. When you route a payment from one channel to another, the blockchain never sees that.