Mr Helms,
Say we want to study how often prime $q$ divides $\prod_{p \leq x}(p-1)$. Maybe write this product as $$ \prod_{i=1}^m\prod_{\substack{p \leq x\\ p \in (q^{i-1}\mathbb{Z}+1)\setminus(q^{i}\mathbb{Z}+1)}}(p-1). $$ If $m$ is the right size relative to $x$, then counting primes $p$ in $(q^{i-1}\mathbb{Z}+1)\setminus(q^{i}\mathbb{Z}+1)$, $i \leq m$ can be done by Dirichlet's theorem on primes in arithmetic progressions.