Great question! I've wondered the same. Harry wouldn't be thrilled and I wonder if Snape could convince him of the urgency (vs Harry trying in GOF the madness of mr crouch). Maybe we'd see Snape cast a Patronus to prove it? But it would be dangerful given Voldemort can't read Harry's mind to know what he knows and is planning. Would Snape stay at Hogwarts if he's not ousted by the Heads of Houses? π€
Long story long, excited to hear thoughts!