Sometimes an abstract of a paper will say something like "we calculate the conductivity of the Hubbard model using Green's functions". Or you ask someone "How did X calculate the spin wave spectrum?" and they will answer, "X used Green's functions techniques." Experimentalists and chemists seem particularly prone to make these kind of statements.
I find this quite unsatisfactory. It is almost as bad as saying "we used a computer" or "we used pencil and paper." Green's functions are just a powerful mathematical technique to simplify calculations in quantum many-body theory. In a sense they have no real physical content.
There are really two key steps (assumptions) in any theoretical calculation involving quantum many-body theory.
1. Choice of a model Hamiltonian.
This could be anything varying from a Hubbard model to a specific density functional in a density functional theory (DFT) based calculation.
2. Choice of an approximation scheme to calculate physical observables (e.g. ground state energy, correlation functions, ...) of the model.
This could be Hartree-Fock, Dynamical Mean-Field Theory, Random Phase Approximation, Second-order perturbation theory, .....
Green's function methods often make it easier (or at least tractable) to do the mathematics of these approximations. But, the technique is not the approximation. The only general case where they can be used without approximation is for non-interacting problems.
When trying to understand a piece of theoretical work these are the two key pieces of information. Try to find them. State them clearly.
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I agree, and your point that exact Green's function techniques are only available in the noninteracting case is very telling. I often get the impression that people tell me they used Green's functions when they don't want to admit first off that they are using a non-interacting model. Only when I press them does this come out. It's like using the phrase "quantum tomography" to blind people to the fact that your model space assumes only single excitons exist...
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