I believe a key point is that the assumption that $F \neq \emptyset$ lets one know that the map $\pi_2: M_G \rightarrow B_G$ has a section, and thus the ring homomorphism $$ \pi_2^*: H^*(B_G) \rightarrow H^*(M_G)$$ is monic.

The way the proof is worded suggests that, in the $E_{\infty}$ page of the Serre spectral sequence computing $H^*(M_G)$, we can then deduce that the $H^*(B_G)$-module generated by $E_{\infty}^{0,*}$ will be a free $H^*(B_G)$--module. [Then the Borel localization theorem part of the argument would kick in.]

This freeness result would be clear if we could conclude that the spectral sequence collapses, but I am not sure I believe that just having that section lets us know this.

I believe a key point is that the assumption that $F \neq \emptyset$ lets one know that the map $\pi_2: M_G \rightarrow B_G$ has a section, and thus the ring homomorphism $$ \pi_2^*: H^*(B_G) \rightarrow H^*(M_G)$$ is monic.

The way the proof is worded suggests that, in the $E_{\infty}$ page of the Serre spectral sequence computing $H^*(M_G)$, we can then deduce that the $H^*(B_G)$-module generated by $E_{\infty}^{0,*}$ will be a free $H^*(B_G)$-module. [Then the Borel localization theorem part of the argument would kick in.]

This freeness result would be clear if we could conclude that the spectral sequence collapses, but I am not sure I believe that just having that section lets us know this.

I believe the key point is that the assumption that $F \neq \emptyset$ lets one know that the map $\pi_2: M_G \rightarrow B_G$ has a section. This then should imply that the spectral sequence collapses, and thus $H^*(M_G)$ is a free $H^*(B_G)$-module. Now recall that $H^*(B_G)$ is a polynomial algebra on $r$ two dimensional classes, and in this ring, any nonzero cyclic module (e.g. the one generated by the class $\pi_1^*(z)$) is free. [Then the Borel localization theorem part of the argument kicks in.]

I believe a key point is that the assumption that $F \neq \emptyset$ lets one know that the map $\pi_2: M_G \rightarrow B_G$ has a section, and thus the ring homomorphism $$ \pi_2^*: H^*(B_G) \rightarrow H^*(M_G)$$ is monic.

The way the proof is worded suggests that, in the $E_{\infty}$ page of the Serre spectral sequence computing $H^*(M_G)$, we can then deduce that the $H^*(B_G)$-module generated by $E_{\infty}^{0,*}$ will be a free $H^*(B_G)$--module. [Then the Borel localization theorem part of the argument would kick in.]

This freeness result would be clear if we could conclude that the spectral sequence collapses, but I am not sure I believe that just having that section lets us know this.