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 now modified as per @alvarezpaiva's comment.)

I know this is trivially true for Zoll surfaces, on which every geodesic is closed. But are there non-Zoll $S$ where geodesics are prevalent enough to yield a positive probability (perhaps $1$)?
          
           _{(Zoll Surface: Image from Polthier&Schmies via this MO question)}