If one would like to drop the Hadamard from the usual $\{CNOT, H, T\}$ gate set, what is a suitable replacement? It seems like $CZ$ might work here but I don't really know how to prove or disprove it.
In general, is there a way to check if a gate set is/is not universal systematically?
CNOT+CZ+T can't be universal because their matrices are all phased permutations (a permutation matrix times a diagonal matrix; a matrix with exactly one non-zero entry per column and per row). Phased permutation matrices are closed under multiplication, so you can't escape into the wider space of matrices. You have no way to create superpositions, or perform interference.
Your gates stabilize the set $G = \text{span}(Z_i)$, and any set of gates that stabilize a finite set cannot be universal.
