Yeah, it is a dark age, but that does not mean everyone died.
The "explosion" that is released from the Relays in the cutscene is the EXACT same as expelled by the citadel, the relays are just a way for the range to be extended.
The "explosion" is shown to have absolutely no harmful effects on organics in the very same scene, let alone going supernova which would destroy every cluster that has a Relay in it, destroying not just some but all of galactic civilization.
Yes it was established that a breach in a working Relay would cause it to go supernova, but this is not a breach, the Relays in the cutscenes expell all their energy, and go completely dark as if off before they break apart.
Bioware shouldnt have to spell that out, if you are intelligent enough to understand most of what goes on in the ME universe, then an implied effect like this should not need to be thoroughly explained. Earth was not destroyed, the Normandy was not destroyed, the planet that Normandy lands on was not destroyed.
The Pillars of Creation which are about 4ly in length a peice were completley destroyed by a supernova. If every relay in the games universe went supernova, almost every single planet and life form (read: all intellegent life) within radius of the relays would be stardust.
That is obviously not what happened.