TLS 1.3 Ciphersuite SHA Level
Firstly, the checksum would seem irrelevant to me given I'm using an AEAD (AES256GCM) cipher. What am I missing here?

Well, what you're missing is what TLS does with the specified hash.  Contrary to your assumption, it does not append a hash to each record; as you point out, that'd be silly.
Instead, it uses it for two purposes:

Within the Key Derivation Function, that is, as a part of the infrastructure that maps P521 shared secrets and public data to actual keys

To summarize the exchange transcript (to detect downgrade attacks)


Which leads us into your next question:

is my risk here (assuming my attacker can find collisions for SHA384 but NOT for SHA512) that my attacker can inject false messages into my communications or could it (very remotely) strengthen my attacker's ability to decrypt my communications?

If the attacker is attempting to listen into a negotiation between two honest parties, he doesn't get to chose what is hashed; that means that a collision attack wouldn't apply (because a collision attack assumes the adversary has the capability to select both values that hash to the same one).  And, even if the adversary were able to inject his chosen plaintext (e.g. via a Javascript applet), that wouldn't help, because what is hashed is selected by the TLS implementations and not the applications.

My question to crypto.SE is- what am I losing in terms of encryption security by having client browsers supporting a max of SHA384?

It wouldn't appear that you're missing much at all; other parts of the system would appear to be more vulnerable.