You have a bad check valve in the carb body. When you operate the primer it is sucking air back through the check valve rather then fuel up through the carb and that is the source of air to pressurize the tank. If a saw has a primer bulb, there will be two check valves in the carb, one will be a nozzle check valve to stop air bleeding back from the high speed circuit into the idle circuit, the other will be a check valve in the body that is just there to allow the primer bulb to function and can be removed with no affect on the saw except that the primer won't work.