Actually, there are also other reasons why one wants to use safe primes in the RSA setting (when working with hidden order groups in cryptographic protocols).
When choosing the RSA modulus $n=pq$ to be the product of safe primes $p=2p'+1$ and $q=2q'+1$, then we also have the following:
The subgroup of $Z_n^*$ of qadratic residues is cyclic and has order $p'q'$. Furthermore, finding a generator of this subgroup is easy, i.e., randomly sample $h$ from $Z_n^*$, then compute $g=h^2$ (which gives us an quadratic residue by definition) and test if $\gcd(g-1,N)=1$ (the latter is proven here). If the test holds, then we have a generator of the subgroup of quadratic residues.
Note that $Z_n^*$ is not cyclic, is of unknown order (if the factorization is unknown) and it is not easy to efficiently sample elements of large order. However, when choosing the setting mentioned above we have a cyclic subgroup of large order and can efficiently sample generators for it.