Ergodicity is a property used to allow the use of ensemble averages in place of time averages. As such, it is only indirectly related to the second law.
In order to provide a statement equivalent to 2nd law, statistical mechanics has to show that the relevant fundamental equation (entropy, Helmholtz free energy, grand potential,..) has got the correct convexity properties. For instance, entropy per particle must be a concave function of its natural variables (energy per particle , volume per particle). Assuming the ergodic hypothesis, statistical mechanics can prove existence and the right convexity of the entropy per particle, under a few general conditions on the interactions and under the condition of dealing with an infinite system (the so called thermodynamic limit). These last conditions are necessary to ensure that the statistical behavior of a mechanical system would reproduce the macroscopic thermodynamics, including the second law..
There is no observable (no function of the microstate) which can be used to define the entropy of a single configuration. Whatever approach can be used, it turns out that entropy depends on all the microstates. So, neither the reversibility of dynamics nor the initial condition is really an issue.