As above, this is a weird one.

When using Schrader valves, the pump chuck pushes the valve core in and holds it open. The bottom of the pump and the tube are all one volume of air until you take the chuck off the valve, otherwise the gauge wouldn't be able to read the pressure. The one-way valve in the pump keeps the air from escaping or pushing the pump piston back out. With a properly working pump you should be able to pump all the air you need with no valve core at all.

Question:

When pumping, does the pressure jump very high in one stroke, then drop back to zero? because that is a symptom of the valve not opening, not a symptom of the valve being stuck open.