Even though other algorithms such as Mergesort also have O(nlogn), Quicksort is normally the preferred implementation because it's relatively easy to do in-place and generally is the most efficient[1] of all the sorting algorithms.
[1]Not my field of expertise though, happy to be told I'm wrong.
This is the beauty of randomized analysis in the worst-cases. The worst-case occurs if and only if the random generator spits out a sorted list. If all permutations are equally likely, a list of n elements has probability 1/n! of coming out sorted = O(n^2).
Even in practice, this is very pessimisstic as it only occurs with a probability of 1/n! and is therefore extremely rare.
To pick that part, first you write your (randomized) QuickSort, then your antagonists picks the worst case, and last you let your algorithm run and pick its random numbers. No matter how the antagonist chooses the input (but not the random numbers), you get O (n log n) expected runtime.
This is different from an average (over all inputs) run time.