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We prove a generalized version of Powers' averaging property that characterizes simplicity of reduced crossed products C(X) \rtimes _\lambda G, where G is a countable discrete group, and X is a compact Hausdorff space which G acts on minimally by homeomorphisms. As a consequence, we generalize results of Hartman and Kalantar on unique stationarity to the state space of C(X) \rtimes _\lambda G and to Kawabe's generalized space of amenable subgroups \operatorname {Sub}_a(X,G). This further lets us generalize a result of the first named author and Kalantar on simplicity of intermediate C*-algebras. We prove that if C(Y) \subseteq C(X) is an inclusion of unital commutative G-C*-algebras with X minimal and C(Y) \rtimes _\lambda G simple, then any intermediate C*-algebra A satisfying C(Y) \rtimes _\lambda G \subseteq A \subseteq C(X) \rtimes _\lambda G is simple. [ABSTRACT FROM AUTHOR]