Does HMAC provide anything that AEAD does not?
If I need to support AEAD sometimes, is it reasonable to just use AEAD all the time and skip HMAC entirely? 


Yes, and if you only  need integrity / authenticity of the message then you can put all of the message in the Associated Data.


  HMAC should be faster, but how much? I’ll need to test this, but suspect the difference might not be huge.


It won't be faster than GMAC generally, but if you include the CTR mode encryption  then you may see a difference. This is too dependent on the system config to be able to answer this reasonably.

Some settings or openssl speed may help you with this. In my case AES-GCM was even faster than HMAC-MD5 which is normally quite fast - although the MD5 speed in OpenSSL isn't that great (but it's the only supported version of HMAC for the speed command, unfortunately). And SHA-256 may also have hardware acceleration. This is a muddy field...


  Is there an advantage to HMAC other than speed and being able to debug data that remains in clear text? Only other thing I know is smaller unpadded payload data.


HMAC is a very sturdy algorithm that doesn't suddenly fail. AES-GCM may fail catastrophically if the authentication tag is too small or if the IV is repeated. I'd say that HMAC is less brittle than AES-GCM: the latter is secure, but it doesn't bend - it breaks.


  Slightly aside, but does HMAC or AEAD benefit from including their nonces in the hashed / additional data? My replay attack prevention will mostly revolve around server time delivered to device on connection included in each message.


No. HMAC doesn't contain a nonce (although you may need a message specific nonce to avoid replay attacks). AES-GCM always includes the nonce automatically during the verification of the authentication tag, so adding it separately is useless.