I think the distinction you're drawing between "the central nervous system" and "the brain" is mistaken here -- the brain is part of the CNS. This kind of reflex basically has to involve the brain b/c it involves both the visual system and the motor system i.e. there's not a fast path from the retina to moving your appendages etc that doesn't include the brain.

The "fully process" part is part of the story though -- e.g. perhaps some reactions use the dorsal stream based on peripheral vision while ventral stream is still waiting on a saccade and focus to get higher resolution foveal signals. But though these different pathways in the brain operate at different speeds, they're both still very much in the brain.