I guess this is a F40 with fuel injection.
low fuel level in VST causing the pump to try pumping air instead of fuel would shut the motor down like you turned off the switch.
lift pump sucking air, tank vent plugged causing a vacuum to form in tank, restricted pick-up tube in tank, return line dumping too much fuel back.
Might monitor the fuel pressure to the VST and rail pressure to give some indication if the problem is fuel related.
did you try pumping the primer bulb to see if that helped any and also may indicate if the VST is running out of fuel if you have tp pump a bunch to get it firm again
