[One Tough Pressure]

When you say pump you mean fuel pump right? There is no reason a failing pressure pump would cause an engine to die. The fuel pump may be going though, or the fuel filter needs replacing.
I had a blockage in my tank that would allow the machine to start and run normal until the blockage would get sucked against the tank outlet, which finally caused the fuel filter to crack from the pump suction. Replaced filter to have it happen again before I realized what caused it to happen.
Also had a bad fuel line from the fuel filter to the fuel pump that got kinked when warm and then choked the fuel supply.