Since it starts it has a spark for at least a minute or two. Unless something is heating or a vibration is shorting or disconnecting a wire, I think we can rule out electrical issues.
To rule out everything fuel related except the carb you could pick up some hose and a few brass fittings. Disconnect the input tube at the carb and rig up a gravity fed fuel supply. Let it fill the carb for a minute or two and then see if the car starts and runs. Use a decent size can or jar to catch whatever fuel is expelled by the pump during that starting and running process.
If the car runs then you know the carb and electrical is OK and the problem is somewhere before fuel reaches the carb. If it quits then immediately start cranking and check that you still have a spark and something heating up is not killing the spark. With the engine not moving at least at an idle speed it might be a tough diagnosis if vibration is the issue but if the spark is OK you are more likely looking at carb issues.
If it ran then on to the possible causes. Packard didn't give fuel delivery specs for 37 but on later fuel pumps which are very similar they did give a spec of 1 pint fuel in 45 seconds. The ambiguity is they did not specify an engine rpm for the measurement but I would assume that would be at normal idle speed. Assuming the engine ran long enough, if the amount of fuel captured during the test is not close to or over 1 pint I would look to a tank, line or pump issue that is starving the carb.
