The Gibbsian ensemble idealizes the notion of repeating an experiment again and again on different systems, not again and again on the same system. So long-term time averages and the ergodic hypothesis, despite the intense interest in them in the first part of the twentieth century, strictly speaking are not relevant to the probability assignment for the state one might find the system in.
However, this changes if there is additional knowledge that the system is being prepared in a particular way some time before the measurement.  One must then consider whether this gives further information which is still relevant at the time of measurement.  The question of how rapidly mixing different properties of the system are then becomes very much of interest.  Information about some degrees of freedom of the combined system may become unusable very quickly; information about other properties of the system may go on being relevant for a considerable time.
If nothing else, the medium and long-run time correlation properties of the system are interesting subjects for experimentation in themselves.  Failure to accurately predict them is a good indicator that relevant macroscopically determinable physics may be missing from the model.