Advantages of a TLS handshake over pre-shared public keys
What are the advantages of a TLS handshake compared to pre-shared
  public keys?


Actually TLS supports raw public keys, so you could actually do what you just described with TLS. Now what it would gain you, for starters, is the fact that you get a protocol which has seen a lot of analysis, which has security reductions, which has some very good implementations and for which  you can somewhat easily / more cheaply buy good security review / consultation on the concrete implementation. 


  transfers it, the AES-encrypted data and an AES-encrypted MAC


For example, TLS doesn't do this (in the handshake), because this is known as MAC-then-Encrypt, which is highly discouraged and has lead to attacks on sloppy implementations in the past. Also TLS has eg measures to prevent replay attacks, ie an attacker just sending your device the same handshake and the same record data again as in a previous session.

Beyond that you don't have to use the full flexibility of TLS. You can just turn off and strip away the code for anything but TLS 1.2 and eg ECDHE with RSA and GCM. This will also have the benefit that you can just use a known-good library and don't have to roll your own protocol and thus save dev-time, nerves and benefit from standardization.