The only bit I don't necessarily agree with is suction due to turning the tap off causing the needle to stick worse - there is a hole in the primer that will let air pressure out.
The suction isn't on the float chamber side, it's on the fuel hose side.
This could happen either way but it's more likely if the fuel hose is empty.
Think of it like this... If the fuel hose is full of fuel because you stopped the mower, and then turned off the fuel tap, and then, after being stored for a week the fuel pressure changes because it's cold in the shed. The pressure and vacuum in the fuel hose will change a bit.
Then instead, imagine if that fuel line was empty and full of air, or partially full of air. It gets cold, the air molecules condense and the pressure lowers, creating vacuum or suction in the hose. It can't suck from the tap side, so it sucks from the needle side, then the rubber tip gets pulled into the seat in the primer cap and gets stuck.
Does that make sense?