In some cases the check valve failing (spring rusting) could indeed cause the pump with its 12V present to not pump any fuel. This occurs when the tank pump has been long dead, the fuel level in the tank is low, the fuel is warm, and the somewhat worn main pump is being asked to suck on a line filled with nothing but vapor to bring the fuel to its positive displacement mechanism. This happens even without a check valve failure.

Getting back on the road with this particular circumstance could be as easy as keeping the tank full with cool fuel until the tank pump can be fixed and the check valve replaced.

Problem is, whoever is actually working on the car gets to say what is wrong and what needs to be done. Not some distant internet volvo enthusiasts.
--
Art Benstein near Baltimore

Question: What do engineers use for birth control?
Answer: Their personality.