It depends in my opinion.

You do mixing to have forward privacy. So noone knows what you use your btc for.

You shall not reuse the addresses that you sent KYC sats onto. Otherwise mixing does not make sense. Because they already know that this address is yours due to KYC, so new sats also will be labeled on this address to you.

If you just generate a new address from same seed of cold storage or generate a new seed for your cold storage both could be an option. I think better to be safe, then sorry. If you generate a new seed, then you don't have any address kyc-ed generated from your seed, so you can't accidentally reuse it.

But you are right, it is not a must.