I was looking at the responses posted on the following question: How is digital signature different from a message authentication code (MAC)?
I read the following comment to the first answer:
So simply speaking, MAC cannot be used to verify sender authentication because it only protects the message integrity. Alright, thank you so much
This triggered a question, what if the key used for the MAC came from an authenticated key agreement?
Both parties would have been able to verify the public keys of each other during the key exchange. So this means that all derived keys from that key agreement are implicitly also authenticated. Meaning that the calculated MAC tags using that key provide message authentication and also provide sender authenticity. And can't seem to find something wrong with this logic.
A second question, is there a security reason why then not to use the public and private key pair to calculate digital signatures on the messages? Instead of using the MAC?