The "proof" is kind of weird. We assume there exists a system that has all three of CAP, but how can we assume that system has the layout in the post with two servers and one client?
For the system to be distributed it must have at least two nodes, and to be available all nodes must respond to requests. So however the rest of the system is shaped, the proof still holds.
In practice you're absolutely right and one approach to distributed systems is to make partition tolerance less important by having lots of redundant links instead.