There are a few hopeful candidates for quantum-secure classical asymmetric key systems, but there are no strong proofs that they are secure against either classical or quantum systems. Without these proofs, and danger that any day someone can come up with a way to break any and all such classical encryption systems, its valuable to at least develop quantum key distribution systems.

In other words, the chances of being hacked are far greater than 1/2^256. They depend on how much secret progress you think organizations such as NSA have made in mathematics. They depend on how likely you think someone will actually develop a quantum resistant asymmetric encryption scheme before large scale quantum computers are built. Of course, there is also a non-zero chance of the quantum internet being hacked.

Yes, developing a quantum communication system will require massive infrastructure. However, its possible to use quantum communication links for regular classical communication i.e.\ once the quantum communication link is high bandwidth enough, you can just route all your classical communication through it. So, its not the worst possible thing to have these links up.