Rather than a secondary objective, how about another obstacle?
It's unlikely that the gens will be slowed down in order to make the games last longer, and I've had this idea mulling around in my head for awhile now, so I figured this was a good place to catalogue it. Doubt it will happen, but it was fun thinking about:
At 3 gens, if there is still 4 survivors in the game, a vaguely angel shaped apparition made from Entity tendrils appears and begins pursuing the Obsession. It moves at half the speed of survivors, however it ignores obstacles and terrain. It has an aura that all survivors can see at anytime. If any survivor enters a 2m radius around The Angel, they get wrapped by it, and it counts as a hook state (similar to PH in that perks don't work with it), and requires other survivors to come save them. Dreadful music and screen effects occur to let them know they're in danger. The Angel can't catch a survivor in the dying state. If the Obsession is being chased or is in the dying/hooked/carried state, The Angel stops in place until the chase ends or the survivor recovers. The Angel disappears when there's only 1 gen remaining, or if the remaining survivors is less than 4.