How tight is the XOR lemma?
Instead of choosing $z\in\{-1,1\}^{2^n}$ uniformly at random, you may want to look instead at more structured (yet "pseudo-random"-ish) functions such as bent functions:


  Definition. A Boolean function $f\colon\{-1,1\}^n\to\{-1,1\}$ is called bent if $|\hat{f}(S)|=2^{-n/2}$ for all $S\subseteq [n]$.


Such functions are known to exist; see, e.g., Chapter 6.3 of Ryan O'Donnell's Analysis of Boolean functions (2014) (or the online version).