What is the point of using statistical mechanics on an integrable system? On one hand, the ergodic hypothesis is usually justified by chaotic dynamics. On the other hand, it seems necessary to consider an integrable system in order to compute a nice, closed-form partition function.
So we either get to relate the ensemble predictions to time averages OR to have a manageable, analytical partition function.
Can we derive experimentally relevant quantities from the partition function of an integrable system?
 A: Let me focus my answer on one particular example of integrable model, since it already allows one to explore most aspects of your question: the Ising model.


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*For this model, you can compute various critical exponents, which are then experimentally accessible, since they are shared by all systems in the same (very rich!) universality class.

*Analyzing "toy models" such as the Ising model also provides a deep understanding of problems that have natural analogues in "real" systems, and share the same qualitative properties. One examples would be the properties of equilibrium crystal shapes (including issues such as wetting or capillarity phenomena, etc.) that can be studied in full mathematical details (these correspond to the Ising model with fixed magnetization, which is equivalent to the lattice gas in the canonical ensemble, and is not amenable to exact computations).

*The important thing is that this model is simple enough (or, more precisely, has a rich enough mathematical structure) to extract relevant information, not that it is integrable. Indeed, it is only integrable in the planar case and in the absence of a magnetic field, while one can derive a huge number of deep properties in the presence of a magnetic field or when the dimension is larger. Among the mass of relevant examples are the rigorous derivation of mean-field exponents above the critical dimension, continuity of the magnetization and sharpness of the phase transition at the critical point in any dimension, sharp asymptotics of correlation functions, etc. None of these works rely on the computation of partition functions.
To conclude, I'd say that this (far too common: it is the way most classes in stat. mech. are presented!) focus on computability of the partition function is a conceptual mistake. It is convenient when available, as it provides a lot of detailed qualitative information. But it is certainly not necessary, and much of what has been derived using exact computations can be derived in other ways, often in a much more general settings where the latter is simply unusable (higher dimension or non-nearest neighbor interactions, more complicated geometries, nonzero magnetic field, fixed magnetization, etc.). 
