Does hashing under the Simple Uniform Hashing Assumption battle worst-case adversaries the same way quick sort does?
Yes, your understanding is correct.  That's exactly the role the hash function plays.

A pair of keys $x,y$ might collide with one hash function, but not with another.  For any fixed set of keys, the expected number of collisions is small.  Moreover, for any fixed set of keys, the probability of having a large number of collisions is very small.  That's why we can say that the average-case running time is good, and why there's only a tiny probability that the running time is high.

To put it another way: Since the adversary doesn't know what hash function we will be using, he can't craft inputs (a set of keys) that will trigger worst-case behavior.  The randomization ensures we only need to worry about average-case behavior (the adversary has no way to force us into the kind of situation that would trigger worst-case behavior).  And, hash tables have excellent average-case running time.