Jerrod,
With the valve set to CITY, there's a secondary channel inside the valve that creates a path between the pump and the fresh tank. You can verify this by holding the valve under a faucet, directing the water into one of the side fittings. Water will come out the opposite side. The same is true when set to NORMAL.
The output of the pump is teed into the water distribution lines going to the faucets. So the pressure of city water can push water back through the tee, back through the pump (assuming the pump's internal check valve is partially open), and into the fresh tank.
When you're set on CITY, the pump should be able to pull water out of the fresh tank and route it to the faucets. If the path were blocked, it would be the same as closing the faucet, and the pump would shut off.
So if on CITY, your pump just runs continuously and never delivers water, there's probably an air leak on the suction side. But since the pump works ok when set to NORMAL, it's confusing.
Have you tried the pump, with 4-way valve set to NORMAL, both with city water connected, and with city water disconnected? And again set to CITY, with city water connected and then disconnected?