And the heart obediently slows down to nothing, at which point it can start to fibrillate and you die. The reason this is rare is dual. Number one, the sensor is small and hard to get to, so it's chance if you happen to be over it. Number two, young people with resilient arteries are almost all resistant to this reaction to carotid compression. Old people with hardened arteries are much more susceptible.
Positional asphyxia is likelier because one tries to get it in judo.
If you're on his diaphragm so he can't breathe, and you choke him, and you hold it too long, he might die. Again, if your choke lasts only a second or three, this is unlikely. It is much likelier if he is hog-tied or even merely face down. I would never practice chokes with a face-down opponent. And you'd have to really flatten the trachea to prevent any air at all from going back and forth.