Security keys very much do have memory and the memory does keep the per website key.

Passkeys do not just allow "password-less" login, much more they all "username-less" login.

If the web site had to send the encrypted private key it would have to know at least the username to send the correct private key.

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Example: "Up to 100 discoverable credentials"

There's your memory.

https://support.yubico.com/hc/en-us/articles/360013656980-YubiKey-5-NFC

What I meant there was security keys before resident credentials. The paragraph after that explains it.