They do darken that fast (not fast enough you can't catch it in a high speed camera, but much faster than a frame). Most of the apparent persistence in the CRT comes from the retina/camera exposure, not the phosphor. A CRT has a sharp peak of light that quickly falls off, but the peak is bright enough that even though it is brief, when averaged out in an exposure in the camera it still appears bright enough to form an image.

You've never taken photos of a CRT, have you? Even at like 400 ISO equivalent, only about a third of the screen is illuminated.

They frequently flicker or are only showing part of an image in video footage or photos, for exactly this reason. They're a right headache to film clearly.

(see this youtube video showing one in slow-motion to get an idea: https://www.youtube.com/watch?v=3BJU2drrtCM)

Thanks for the link. It seems i'd concluded this a bit wrong from seeing those half lit frames (like at 1:35 of this YT video).