As you pointed out correctly ergodic measures are extremes of the convex set formed by invariant (w.r.t. to the semigroup) measures. As such, if you can show uniqueness of invariant measure then the result you are looking for will follow immediately by Birkhoff's ergodic theorem as you point out.

Uniqueness of invariant measure for markov processes usually follows by Harris theorem ( see for example, "Yet another look at Harris' ergodic theorem for Markov chains" by Martin Hairer and Jonathan Mattingly ) which depends on specific properties of the semi-group and not necessarily on the path wise properties of the process itself.

As you pointed out correctly ergodic measures are extremes of the convex set formed by invariant (w.r.t. to the semigroup) measures. As such, if you can show uniqueness of invariant measure then the result you are looking for will follow immediately by Birkhoff's ergodic theorem as you point out.

Uniqueness of invariant measure for Markov processes usually follows by Harris theorem ( see for example, "Yet another look at Harris' ergodic theorem for Markov chains" by Martin Hairer and Jonathan Mattingly ) which depends on specific properties of the semi-group and not necessarily on the path wise properties of the process itself.