I expect that thing that looks like a Schrader valve is snifter valve. It was there, as you thought, to admit air, but only down maybe 10 ft. At that point is a device that I forget the name of, but it should let water out to let the pipe take up several feet of air. If the water pressure at that point is low, that thing lets out more water. This was used with the old style pressure tank that had no diaphragm or bladder.
At this point your action, assuming you now have a diaphragm pressure tank, is to remove the snifter valve or somehow cap it. I don't know if its thread is compatible with a normal valve cap. Also remove the topside checkvalve.
If the pump cycles significantly due to that gadget leaking, you could have that gadget removed. Ideally it will not pass significant water, because there is pressure there after the checkvalve is gone.
Edit: Whoops. I missed the " galv tank" part of your posting. If you wanted to put in a diaphragm pressure tank then you would remove what would become the extra stuff.