I think you're confusing the SHA-2 hash core with a MAC derived from SHA-2. MACs and hashes aren't the same thing. You can build a MAC on a hash --- in fact, you can build a MAC on a hash that would be terribly insecure on its own, and some of those MACs are state-of-the art; see, for instance, all the polynomial MACs.
That's what I'm trying to articulate. Hashes aren't MACs.
This is partially my fault, for playing loose with SHA-2 512/256 vs "The Prefix MAC built on SHA-2 512/256".
Again: I think we should just call that SHA-2-MAC. It's an unambiguous name that couldn't mean anything other than "truncated SHA-2 512 used as a prefix MAC".
It's hard enough to follow this topic for many people that want to keep updated. The fact that you have to spell out "please use SHA2-512/256, but not SHA2-512 and not SHA2-256" is mind-numbing.
"It's really SHA2-512/t where t has a value of 256 and that means that it's SHA2-512 truncated down which alleviates length extension attacks." Just typing that sentence gave me a headache and I have no real idea if it's right or not.
Adoption would be easy if communication was easy. As it is, it's a PITA to even have a short conversation on the subject.