No you are correct, because modulo should be prime or at least coprime to base, unless you want to generate random prime

It makes way more sense intuitively to use different random coprime modulos. Consider that as long as base > |alphabet|, a polynomial hash without mod will always be a unique number. If two strings have these hashes $$$h_1$$$ and $$$h_2$$$, then a collision modulo $$$m$$$ means $$$h_1-h_2$$$ is divisible by $$$m$$$. When you use several coprime modulos $$$m_1, m_2, \ldots$$$, a collision means that $$$m_1 \cdot m_2 \cdot \ldots$$$ divides $$$h_1-h_2$$$. I don't know how it works with random bases, but with random modulos, as long as the polynomial hashes give sufficiently random remainders modulo $$$m_1, m_2, \ldots$$$, the chance of a collision drops extremely quickly with added modulos.

You have a good point: If you only worry about system tests and not about hacks, then I could see that a base smaller than the alphabet size is more likely to cause collisions. If we model system tests as uniformly random strings of various lengths, then for bases larger than the alphabet size, small strings won't collide and large strings will have approximately uniformly distributed hash values, so chance of collision might be around $$$\frac{1}{mod_1}$$$. Bases smaller than the alphabet size will have many collision on small strings, so they're worse.

Let's compute how unlucky you'd have to get: With $$$mod_1 = 10^9+7$$$, the probability of getting base below $$$256$$$ is only $$$2.5 \cdot 10^{-7}$$$. If you're comparing $$$10^5$$$ pairs of strings, then the probability of getting a random collision on random testdata is already $$$10^{-4}$$$, which is much higher. So I'd worry more about random collisions that about getting a small base.

Ideally you use multiple hashes with large modulus so that you're incredibly unlikely to get a collision even with hacks. Then you'll have an implementation for which you can prove that it will work on any input with high probability (and don't need to hand-wave about how you model system tests). For instance, if you use $$$4$$$ hashes with $$$mod = 10^9+7$$$ and randomized bases, $$$n = 10^5$$$, then the probability of a single comparison colliding is at most $$$\left(\frac{n}{mod}\right)^{4} \approx 10^{-16}$$$. If you're comparing $$$10^5$$$ strings, then the chance of a collision is still only $$$10^{-11}$$$ per test case. (If you use $$$mod=2^{61}-1$$$, then the probabilities are even better, but the implementation becomes tricky.)