Why can't you just use a prefix MAC with SHA-2 512/224?
You can, and you just invented the Sponge construction in the process! :)
The key idea is to only expose part of the state to inputs/outputs.
Sure, but SHA-2 512/224 predates SHA-3 and the Sponge construction by, what, a decade and a half?
And prefix MACs predate all these things by multiple decades (they just happen to be insecure in full-width-output MD hashes).
I don't think the ability to use Keccak as a simple prefix MAC is all that gamechanging. You can hash data once --- and much, much faster --- by using SHA-2.
Keccak allows for very simple and efficient implementations of all these primitives.
> H(key ‖ m)
This is insecure for SHA2 (length extension). However, it is secure when done with SHA3. In fact, it's pretty much how you construct a MAC from it, that's how natural it is.
HMAC was invented specifically to deal with the shortcomings of SHA2 and predecessors. For reference, it is H(key ⊕ opad, H(key ⊕ ipad, m)), where opad = 0x5C… and ipad = 0x36…. Compare that with the above and you'll see what I mean with simple and efficient.