Not only self-aggrandizing but the implication is that the rest of the field is limited or struggling in some way. If people haven't realized some connection, just say that.

Very good. We all need something to believe in.

(clearly) outgrown

You're welcome! Sorry for it I set off on a bit of a rant! Really there's a lot more to the formalism behind a physicists' derivation or DMFT than what physicists describe, and mathematicians ignore. I'll have to share some material with you at some point!

Though the latter show for non-Gaussian disorder they can't get LDPs but still get "universality". I don't know what the implications are in those cases for the DMFT equations' accuracy / "exactness"!

The justification for studying the disorder average /unconditional for the spin glass & NN case, would appear to be large deviations principles, a la Guionnet and Ben Arous ~1996, and more recently stuff from Dembo, Zeitouni & collaborators.

Actually, what they're considering is the unconditional distribution of some state trajectory x_0,...x_T, where the quenched disorder one is conditional on a realisation of the disorder. I never know why people don't just say that.

Disorder muddies the water, so to speak, because the DMFT most people in machine learning encounter (e.g. NNs , and AMP I believe) is one for the quenched disorder system which physicists study by considering the "disorder averaged generating function".

e.g. loopwise expansions, of which the Hartree-Fock approximations is the most elementary mean field approximation. Helias and Dahmen do a pretty good job of walking through the field theory language for SDEs, ditto Chow and Buice, and Hertz, Roudi and Sollich.

In physicists' language the spirit of a MF theory is to replace an interacting system with a non-interacting one, though each particle or site "feels" an effective field.
The DMFT for strongly correlated materials is definitely in the quantum domain, and so uses field theory terms.

Yep I think the issue is there's a grab bag of various derivations and from there different departure points for different approximations, but the fundamental ideas are still the same: approximate a distribution via a simpler one, typically full factorisation with a dependence on the "mean field".

yep that's the one. What's the other DMFT? the one with the wikipedia page? I think they're probably more related than I used to think.
What's your reason for looking into it? I have a few resources I can happily point you to, either more rigorous direction or with better physics pedagogy.

For reference, which DMFT are you trying to get into?