Inside the TPM would be a great place to store the private key, simply because it cannot be extracted.

Of course, that introduces a new problem: securely backing up the private key.

Doesn't most computers have TMP chips, which are perfectly capable of storing and generating private keys so the browser doesn't have acccess to them.

Potentially, yes. But now we're back to technology/software which doesn't yet exist. If you create it I would use it, though.