But the sender (presumably who sells you btc) must know about your channel. If it is private, you will send him a routing hint.

This could be avoided if you open a parallel public channel to the same LSP.

Also, force closure may reveal that it is a channel. Not sure about the taproot channels in this context.

Finally, at least lnd (so Blixt too, not sure about other implementations) may happily sweep and combine various UTXOs after forced closure, this also leaks some information.

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Routing hint will only contain scid alias. not the channel ID tied to the utxo.

yeah, only a force closure will reveal.

ok, makes sense. You still reveal the fact that there is private channel between you and LSP, but with taproot channels good luck finding the channel UTXO.

what is the persistence of this information?

assuming no force closures, is the data "gone" after you burn the channel?