A. Neumaier said:
It is time invariant hence stationary but leads to completely wrong predictions for thermal q-expectations such as the internal energy.
How do you come to this conclusion? It depends on the system!
Of course the MEM doesn't tell you what's the relevant information for a given system. That you have to determine yourself. The MEM is a very general principle. It's almost inevitable to make "objective" guesses based on the available information about the system.
Now 2 issues are usually raised against it, also in this thread.
(1) Choice of the "relevant observables" to be constraint in maximizing the entropy
Your criticism is quite common: Given only the constraint that you want a stationary state, constrains the possible choice of "relevant information" to be either averages of conserved quantities (like for energy in the canonical and grand-canonical ensemble; in the latter case you also give the average on one or seveal conserved charge-like quantities) or strict constraints of conservation laws (energy is strictly in an on macroscopic scales very small interval, as for the microcanonical ensemble).
That usually the Gibbs ensembles are preferred, i.e., giving at least constraints on ##U=\langle \hat{H} \rangle## and not some on any other function of ##f(\hat{H})##, is due to the fact that one considers the thermodynamical state as a small system coupled (weakly) to some reservoir, defining the canonical and grand-canonical ensembles (depending on whether you allow for exchange of conserved charges or not). Treating than the closed system large (reservoir+system)-system as a microcanonical ensemble (where it doesn't matter, which function ##f(\hat{H})## you look at), you are inevitably let to the usual canonical or grand-canonical ensemble with ##\langle \hat{H}_{\text{sys}} \rangle## is the relevant constraint. In the thermodynamic limit higher cumulants don't play a role. For small systems it can be necessary to take such higher-order constraints into account. The same argument holds if you have other relevant conserved quantities to take into account: In the typical "reservoir situation" for macroscopic thermodynamical systems the additive conserved quantities are the relevant observables and usually not more general functions thereof. For details, see
https://journals.aps.org/rmp/abstract/10.1103/RevModPhys.85.1115
and references therein.
This view is also solidified by the usual dynamical arguments using non-equlibrium descriptions of open quantum systems (master equations, transport equations, etc.): For short-range interactions in the collision term and truncating the BBGKY hierarchy (or the corresponding analogs for more detailed quantum descriptions; as the general Schwinger-Dyson hierarchy of QFT) at the lowest order leads to the standard equilibrium distributions (Bose-Einstein, Fermi-Dirac, Maxwell-Boltzmann), corresponding to the choice of additive conserved quantities in the MEM as discussed above.
(2) Choice of the information measure
This is the question, which type of entropy to use. The Shannon-Jaynes one in the physics context refers to the classical Boltzmann-Gibbs entropy. It's well-known that this doesn't work with lang-range forces present (electromagnetic, but that's pretty harmless, because in many-body systems you usually have Debye screening which comes to the rescue, but there's no such thing for gravity, and that's important for structure formation in the universe and our very existence). Here other (non-additive) entropy forms like Renyi or Tsallis entropies may be the better choice. As far as I know, there's however no generally valid dynamical argument as in the Boltzmann-Gibbs case.