Which bit might be stiff? The valve itself, the part that moves might be stiff. The valve might not seem impressive but it is a little machine and any moving part that does not get lubricated can get stiff - whatever you make it from.
The core (removable or not) has a moving part in it; when you push on the pump it creates an area outside the tyre with a higher pressure than the area inside the tyre, the valve is pushed open and air flows in. When the pump is removed it is the air pressure that holds the valve closed - we screw them up tight to prevent slow leakage(such as it being knocked open ever time you go over bump). This is different to some other types of valve that have a spring keeping the valve shut & so don't need to be tightened manually.
It sounds like you gave it bit more umpfh - so making greater pressure differential and overcoming whatever stickiness was in the valve. It might have had a bit of debris in there, or have been bent very slightly or it might have been something else.
bree - unless you go very very cheap I suspect almost all track pumps are ok. The ones up to £20 or £25 are pretty much disposable. The nice steel ones that cost more have spare parts available and can be rebuilt. Mine is a Blackburn Air Tower 2 - I bought it because it was the one the LBS handed me when I said I wanted to buy a track pump. It was about £20 if I remember, or maybe £15. When it eventually breaks I might get one of the Lezyne ones but that is because I covet the nice shiny objects not because it does a better job than an ordinary one.