Chat delay does not affect e.g. command block feedback (seen in 1.16.2 Singleplayer). And on a modded server, not even chat is necessarily affected (seen on a 1.16.1 Bukkit server).
I would expect chat delay to be a client-side feature that simply buffers everything, but apparently it also requires some cooperation from the server for some reason (which is broken for 1.16.1 Bukkit and either broken or just not enabled for command block feedback in Vanilla).