Perhaps this is not the strongest result one can get, but it is true that for almost all $p^*\in M$, the set $F_{p^*}$ you define in the question has measure $0$. This is equivalent by Fubini applied to the product measure space $M\times M$ to saying that for almost all points $p\in M$, the set $\{p^*\in M;p\in F_{p^*}\}$ has measure $0$.
That is, it will be enough to prove that for all $p\in M$, the set of points $p^*$ whose distance to $O_p$ is minimized at more than $1$ point has measure $0$. But $O_p$ is closed for all $p$, because the action of $G$ on $M$ is proper, so by the answer I gave to your other question, we are done.