Let $S$ be a closed surface embedded in $\mathbb{R}^3$, let's say of genus zero. I seek examples of $S$ with the following property: If one selects a random any point $p$ on $S$, and a random direction $u$ tangent to $S$ at $p$, then the geodesic issuing from $p$ in direction $u$ is a closed geodesic with positive probability. [Question modified to reflect @alvarezpaiva's comment.]

I know this is trivially true for Zoll surfaces, on which every geodesic is closed; see figure below. But are there non-Zoll $S$ where geodesics are prevalent enough to yield a positive probability (perhaps $1$)? Or, to be more explicit:

Q. If, for every $p \in S$, closed geodesics issuing from $p$ occur with probablility $1$ among all tangent directions $u$, does that imply that every such geodesic is closed?