A friend has a '88 VW Fox, which had a similar problem: “Sometimes it won’t start. Its not the fuel pump. It has already been replaced”, he said. The problems would only occur if the tank was less than half full. I agreed to help him repair it.
Turns out the VW Fox has two fuel pumps: One inside of the tank and a second one mounted next to the tank for generating the high pressure needed for the injectors. The pump inside the tank feeds the second pump.
We disconnected the fuel line outside of the gas tank and measured the fuel output rate if the first pump. Zero.
The pump inside the tank had failed, and the output hose of this pump had rotted out within the fuel tank. If the tank was full there was sufficient pressure for the outside pump to get gas. But once the fuel level fell below the rotten hose the outside pump got air, causing the engine to run rough and to not start once stopped.
Replacing the pump and the hose solved the problem.
Some cars have multiple fuel pumps, check all of them, and check for a bad suction hose within the fuel tank.