Speaking from my own experience today, no. None of my characters have done the starter quest for the event. I got event tickets, a gold coffer, and two daily coffers upon turning in the first daily I did. I got a purple coffer and two daily coffers for each of the three other available dailies. I also picked up a couple of other purple coffers from looting various stuff along the way to/from the daily locations. Everything seems to be working as intended.

EDIT: the major event rewards are tied to the delve/WB dailies in Blackwood and The Deadlands, not to the delve bosses and World Bosses. Those are still own thing when it comes to loot and are not tied to the event.