The coordinate ring of an affine algebraic action is a locally finite rational module

F1 makes $A=\mathbb{C}[X]$ a right comodule whose evaluated representation is exactly $f\mapsto f\circ g^{-1}$. Apply F2 to any finite subset of $A$ to obtain the required finite-dimensional stable subspace with algebraic $G$-action. By the definition in Classical complex affine algebraic actions and rational modules, this is rationality and local finiteness.

Since $c$ is an algebra map, evaluating its second factor at any $g$ gives $(g(fh))(x)=f(g^{-1}x)h(g^{-1}x)$ and $g1=1$. Thus each $g$ acts by an algebra automorphism, with inverse $g^{-1}$. The hypotheses of F1 and F2 cover reducible and empty $X$ and disconnected $G$, so none of the excluded extra hypotheses is needed. AC is inherited solely from F1's affine reconstruction route.