I don't know about progress. We need to account for new players and can't have Guardians cleaning up the moon and there's nothing left. But the Guardian public event sounds sick. It would really make you feel like you're a part of something. They could beam down from their jump ships.
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There is no reason that once we cleaned up one area, that the enemy couldn't enact a counter offensive and regain control of that area or a new area. It could be a weekly thing. Progress could be determined by total number of enemies killed by player characters. We know they can track that stuff. There could be a goal for completion, like 20,000 hive killed by all PCs or something and rewards for level of participation to each player. Remember the ME3 Galaxy at War Map? Yeah, it can be done.
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They're not "cleaning up the moon" they've merely created a new outpost, a way for guardians to stop and rest on the moon and restock or turn in things. It just makes the environment more alive and interact able.