I wonder if the probability is dependent only on $r_i$, or also dependent on the placement of $R$ within $S$? In these two examples,
it takes an average of $2.70$ steps to reach slicing $R$ on the left, but $3.16$ steps on the right.
I realize I'm ignoring your condition that $s_i \gg r_i$.
**Added 4Aug2020**. I include below some simulation data that might help a theoretical investigation. Here are two examples where $R = 0.2 \times 0.1$ in a unit square $S$.
On the left, after one million trials, the probability that the long side of $R$ is sliced was $0.591$. On the right, the probability was $0.622$.