Recently Lubetzky and Peres showed that simple random walks on a sequence of $d$-regular Ramanujan graphs $G_n=(V_n,E_n)$ of increasing sizes exhibit cutoff in total variation around the diameter lower bound $\frac{d}{d-2}\log_{d-1}|V_n| $. We provide a different argument under the assumption that for some $r(n) \gg 1$ the maximal number of simple cycles in a ball of radius $r(n)$ in $G_n$ is uniformly bounded in $n$.