I agree . . . no air or water either means the port is clogged (possible, but not likely if membrane is OK), or all air is lost. If dry and no air, that Schraeder core is the only seal that can have let it out . . . might consider changing the core while you are at it, or you may lose air again. And back it with a good metal cap with a rubber seal in it. That setup is used in AC systems high and low, up to 450psi or so . . . but the cores are more for service reasons and not a primary seal.
Not sure on your pressure . . . my home with a we'll runs more like 40 to 45, and not sure on the one with city water. No PX proper with city, but they might have put it equivalent with pipe stubs in tne wall upstairs . . . . just know we have never has a problem.