Let me help all mages out there as I fell victim of the same thing and searched all over. Someone mentioned about joining a different instance and then leaving the instance and walking up the stairs.
Well that seems to be the temp fix for this quest. So, here's what you will do. Before heading up the stairs to Archmage Vargoth, join a random lfg like... Levantus, this is definitely the best way to jump you into another instance. Once you get autoaccepted, kill the mobs that reappeared, and then leave the lfg party and walk up the stairs and it will trigger the rest of the quest for you to complete.
I explained it a little bit better here: https://youtu.be/X9OBZvBVKrU
Credit: http://eu.battle.net/forums/en/wow/topic/17613342003
Mondan - Dragonblight
EDIT: So I abandoned it a couple more times, joined a random party from LFG tool (which didn't allow the parts of the quest to work) and then left the party and carried on with the quest then walked up the stairs and it had fixed. Not sure if that will help you much but that is what I did.
EDIT 4/10/17 - Some people are still stating that this issue is known and are having issues with hopping to another "instance." I know I kept saying instance, wrong terminology, my apologies.
USE THIS TOOL - https://mods.curse.com/addons/wow/crossrealmassist
Cross realm assist automatically does a group finder for you and places you into a different realm. Then you leave the raid/group that you found and continue on the quest.
"Happy Questing!" And Aluneth is still a sassy staff. ;3