The specific weakness of these systems is that the governments cannot be trusted with this. They have demonstrated this - Snowden leaks for the US and several EU states, and the general monitoring clause in the Data Retention Directive for the EU as a whole.

Even if these governments come up with a zero knowledge system, it's only one click away from being replaced with a full-knowledge system, because the user is already used to it. These governments have already tried spying on everyone (and they almost certainly still do).