MAC providing sender authentication (follow up question)
The statement you copied from the other question is incorrect. A MAC does authenticate a message (would be a rather stupid name, otherwise). If you have verified a MAC, you can be sure that the message was sent by anyone knowing the secret key. This is slightly weaker than the authentication done by a digital signature, because there are usually more parties in possession of the key.


  This triggered a question, what if the key used for the MAC came from
  an authenticated key agreement?
  [...]


Your logic is correct. That is exactly what most secure network protocols (like TLS or SSH) do.


  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?


That has performance reasons. Public Key operations are slower, need more memory and the output size is larger, compared to their symmetric counterparts.