I don't think this is true for $G = \mathbb{Z}$. Let $X$ be a singleton and let $\alpha$ be the trivial action of $\mathbb{Z}$. Then $C(X)\rtimes \mathbb{Z} \cong C^*(\mathbb{Z}) \cong C(\mathbb{T})$, which is abelian so it has lots of tracial states, but there is only one probability measure on $X$.