Is Wikipedia's use of "quasi" in the title for the enzyme kinetics reflective of [...] the further assumption that $[S]$ is constant[...]?
No I don't think the "quasi-steady state" hypothesis is related to whether $[S]$ varies with time.
[W]hy add a further descriptor that denotes uncertainty or inaccuracy when we already know that steady state is an assumption?
Different fields use different terms for the same concept. For example, some folks call it the "pseudo-steady state hypothesis" and abbreviate it "PSSH". Some people just say "steady state assumption". It's all different vocabulary for the same thing.
I don't think there's a principled reason to prefer one term over another. If you solve the full differential equations for kinetic models that would normally invoke the PSSH, you'll see one problem. Before the reaction starts, the concentration of intermediate $I$ is zero. But when the reaction is going on, we assume the intermediate is at steady state, i.e. that $\frac{dI}{dt}=0$. But if $I$ starts at zero and has an initial concentration of zero, then it is everywhere zero, which violates the prinicple that intermediates actually occur in real-life chemical reactions. So there are two phases to such models: the first is the "burst" phase, also known as single-turnover conditions, or unsteady state kinetics. In that phase $\frac{dI}{dt}$ is not 0 but increases rapidly. That phase ends quickly and is replaced by the pseudo-steady phase for $I$. If you want to get precise about the strict definitions of the two phases, you'll need to invoke singular perturbation and asymptotic matching, as described in this thorough book chapter from Mathematical Biology by James D. Murray.
So you could argue that the "pseudo" or "quasi" is good to have in there to remind you of these complexities. Or you could argue that all those extra letters aren't doing a whole lot of good. Either way, it's just a semantic issue and is unrelated to differences in enzyme kinetics and more "traditional" chemical reactions.
