Is this protocol design secure?
See below.
What would be the advantage to use HMAC (or any other MAC) instead of AES encryption?
HMAC was originally proposed as construct that turns a Merkle-Damgaard hash function based on compression functions built from block ciphers. Although there's no decryption in HMAC, you can actually easily replace your AES decryption-based authentication with comparing HMAC-signed tag.
Did I miss something?
What you're doing is essentially an authenticated security transport without confidentiality.
What you're missing is that you didn't authenticate the exchange following the initial authentication - this allows for connection hijacking, arbitrary injection of data packets, and more.
Therefore, you should use HMAC+[hash] or CMAC+[block cipher] to authenticate all of your messages exchanged, and also keep a counter as a state to detect duplicate packets and message replay attackes.