This is not a software engineering problem but social and political one. Thus, any attempt to solve it only by means of some smart engineering would be bound to be rendered uneffective for most or all of the masses.
Let's suppose government allocated $200 billion to create this infra. But there are free tools which can render it useless. They are less likely to go ahead with such a plan then.
Whole premise of making inroads in service providers servers is that information stored there is human readable. Store it encrypted. Come up with stronger algorithms and security schemes and this problem is affectively solved.
If we can reach to a stage where only option remain to read a person's information is to arrest them or break into their devices, we have affectively pulled a project like PRISM down.
Nobody is inroading, government has access to the data by law or by policy. Service providers, that operate under the jurisdiction of a state, cannot perform actions that do not adhere to the laws and policies of that state. It's as simple as that.
> Come up with stronger algorithms and security schemes and this problem is affectively solved.
Algorithms and schemes are already there, but there's no legal and/or policy framework to force their use. In fact, the current framework is designed to force the opposite.
I agree with your point about existence of such algorithms. But we can certainly improvise. What you have written here is like saying that Facebook is just a message board. Of course, technologies exists. It's about how to utilize them and make them easily accessible. And again, there's no framework to force any communication channel. There's no law that data stored on servers cannot be encrypted.
If it is lawful then there's no reason for Snowden to run and hide. In fact you are insinuating that what he has done is unlawful.
Still that doesn't solve the the backdoor issue. This one is much harder to solve.