They still hold a copy of their own private key somewhere, you're just punting the issue a little bit. Plus, there would have to be a single key for all users,

They still hold a copy of their own private key somewhere, you're just punting the issue a little bit. Plus, there would have to be a single key for all users, or you'd have to give every user's key to the institution as well. That means more travel over the wire, that means central storage of skeleton keys, etc. Each of these factors introduces another vector of possible attack. If there's a gate, there's a way to get in, and no matter how many keys are required or where they're kept, they'll always be more vulnerable than a wall with no gate.