The first time I ran through the Great Escape event in Neutropolis, I had one teammate and we were the only players who participated in the event. We formed our team before the event started and were waiting at the event location before it began. At no time during the event did I disconnect or leave the team. My teammate, who had done the event once before and had the badge, was offered the large rewards table with the emotes, but I was offered neither badges nor emotes.
After having to run not just this event but both the other Praetorian events twice to get the larger rewards table, I put in a /petition to figure out if something was up. I was told that my "participation level didn't meet the requirement for the top-tier rewards and so they weren't offered. When you ran each of these a second time, your participation level was higher than before and, as such, you were offered the highest tier of rewards for your efforts. This is working as intended."
I asked what the standards for "participation level" were, but the GM was unable to provide a response as "That is a question of design and, as such, is best asked of the Development Team through the City of Heroes forums."
Damage, in itself, is not part of the reward criteria.