Again, maybe someone more knowledgeable can explain more - I'm a consumer of TPMs, and I understand the concept, but I'm no expert.
That doesn't come with any particular privacy concerns however.
But web is really not so much my thing, so I'm now at the point where I'm probably going to start saying incorrect things.
So the question is how a particular protocol works, and whether a site can still identify you after you've deleted your session cookie. As far as I understand it, if you were wantonly using webauthn for all your session cookies, it would be relatively easy for a site to check if you're the same person as before.
Nope, the thing is that you ask the TPM to sign something with the key/secret, but the TPM will not allow you by design to exfiltrate the secret.
Separately from that, using the TPM to lock the session cookie to the device sounds useful.