I've read the manual and I know how sustain mode is supposed to work but as you can see in the video the Prophet-5 is not responding to the sustain pedal when it's in sustain mode. It is working when I switch it to "hold" so the problem is probably not the sustain pedal.
Everything your video shows is correct behavior. When your Prophet 5's Release/Sustain global is REL (the original Prophet 5 behavior), there is no visual feedback of any kind. When the Release/Hold button light is on, the envelope naturally moves through its release stage. When the light is off, the envelope will skip the Release stage completely UNLESS the pedal is held down. When the light is on in REL, the pedal does nothing.
So try this:
(1) Enable REL in the Release/Sustain Global
(2) Find a sound with a long release, or just crank the Release knob all the way up.
(3) Turn off the Release/Hold light
(4) Play a note without sustain pedal down. Hear how the sound just cuts off?
(5) Play a note with sustain pedal down. Hear how the release stage now works?
(6) Turn on the Release/Hold light. The behavior is now the same as when the pedal is down, and the pedal does exactly nothing.
I personally prefer the more modern sustain pedal behavior of "pedal holds the sustain level and the sound moves to the release stage when the pedal is lifted." So I just leave my Release/Sustain setting at HLD. It's more piano-like, I think.