Can I think that the many-body Green's function method is just a
The answer is definitely NO.
I have also used the above-mentioned Nolting's book to learn the basics of quantum field theory for condensed matter. In general, one should be always very careful when an intricate method or principle is described in loose wordings (just to serve as an introduction). If you continue your study towards the chapters where Nolting introduces perturbation theory, you will realize the meaning of those introductory sentences much better.
What he refers to in this sentence, “Its basic idea consists of replacing an inherently complex interacting many-body system by a free gas of quasi-particles”, refers, first and foremost, to the concept of Fermi liquid theory. Crudely speaking, a Fermi liquid is a Coulomb-interacting gas of electrons, where -- according to Landau's theory -- each electron of the original non-interacting gas is “screened” by a “cloud” of accompanying electrons (and holes), so that, effectively, this cloud can be considered, within a certain approximation, as a fermionic particle itself (just like the original electron); for a nice pictorial introduction, see Mattuck, "A guide to Feynman diagrams in the many-body problem" (WCat). Hence, in this way, the original complex strongly-interacting system of fermions can be considered (within the bounds of the approximation) as a system of weakly-interacting (or quasi-free) fermion-like particles (called fermionic "quasi-particles"). Therefore, the complexity is reduced and a perturbative treatment via Green-function techniques becomes feasible. For a detailed exposition of the concept of Fermi liquid theory, refer to Abrikosov, Gorkov and Dzyaloshinski, "Methods of quantum field theory in statistical physics" (WCat).
Remember that in general, the success of this Fermi-liquid approach in prediction of the real physics of an interacting system is not guaranteed. There are indeed many cases of high interest where that quasi-particle picture breaks down completely leading to novel phases of matter -- a well-known example is superconductivity; see also non-Fermi liquids.
Standard mean-field theory is just one of the lowest approximations one might perform to obtain the properties of an interacting system. There is no guarantee that it will suffice, and in many cases, it miserably fails. However, it is nonetheless important to obtain the first insights, and as a benchmark for more advanced approximations. As a side remark, a mean-field approximation can be formulated elegantly in terms of the Green's functions.