> 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 a construct that turns a Merkle-Damgaard hash function based on compression functions built from block ciphers, into a message authentication code. 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.