Weakness in Pohlig-Hellman algorithm
The point is that even though $p - 1 = tq$ may be large, the discrete log security of $(\mathbb Z/p\mathbb Z)^\times$ against Pohlig–Hellman depends on the size of $q$, not on the (possibly much larger) size of $p$ or $tq$.  If $q$ is the largest prime factor, then the cost of computing discrete logs modulo $p$ is essentially at most the cost of computing order-$q$ discrete logs in Pohlig–Hellman.  This is why, e.g., Schnorr groups are chosen to have an order with a large prime factor.