Does the future SIGHASH_ANYPREVOUT capability hurt the privacy of Taproot lightning channels?
The foundation of a channel is a multisig output which can't be distinguished from other outputs when using musig and taproot.
The mutual channel close is just a regular tx spending this output so I'd say it'll be hard to detect this as a lightning Chanel without other data (similar as with current channels)
However the unilateral close is different and leaves a footprint of the channel state. I think the update transaction that is used to kick of the settlement tx might have the detectable structure that you have been referring to. (I'd actually like to ask you for your source)