My server logs are still full of "Player moved too quickly" errors after using Waystones and other teleportation devices, and sometimes simply from flying with an Elytra. I thought this mod was supposed to prevent that check?

It does not prevent it completely. (Just had a quick glance at the mixin for player speed limits)
They have simply raised the limits for what triggers it by default, but it also seems to be configureable.
If there's some assfuckery within Minecraft itself so that speed is not calculated correctly and fast enough, it might still think you moved at the speed of light or whatever (lag can also cause this).
This assfuckery can also come from using other mods.