The Breen might have the ability to destroy the Archive, but with Discovery gone, it wouldn’t gain them anything and how ever long it took them would be time lost for chasing Discovery.
That's a hell of a gamble if your objective is to protect civilians (and likely a gamble they would lose, considering the Breen's actions).
why didn’t Discovery prepare a fake clue?
I think the answer is the same - it's not something you want to screw around with if you're trying to save lives.