Symmetric encryption is no longer necessary, because all security services can be implemented with public-key cryptography.
I suppose that is true enough at a high level; if all you had was public key cryptography, you could do symmetric cryptography (if necessary, giving both sides copies of the private and the public keys); this would meet the security requirements against adversaries that don't have the keys.
On the other hand, this skips a rather important point about symmetric key cryptography; efficiency. Symmetric cryptography is thousands of times faster at encrypting and authenticating data. In practice, public key cryptography is almost never used alone; if you encrypt a message, the most common approach is to pick a random symmetric key, use that to encrypt the message, and then use your public key encryption method to encrypt the symmetic key. Similarly, if you sign a message, you generally use a symmetric cryptographical hash to summarize the message into a short hash, and then perform the signature operation on the hash. Yes, you could do everything using only public key primitives, but no one does.
Moreover, in public key cryptography the operations for authentication and confidentiality are exactly the same.
Nope; public key encryption does not provide any authentication (as anyone with the public key can generate the message), and signatures do not provide any confidentiality.
If I were to speculate what the author was thinking about, I would guess he was thinking about RSA; with RSA, the confidentiality (public key encryption) and the authentication (signature) method are distinct (the padding methods are different), but they do rhyme.