This is actually hinted at in an existing answer (on re-reading) but (and this is a rather handwaving take on the question as we don't have much data on tyre deformation, expanded from an intended comment):
There's a further explanation which I believe to be closer to the correct one, as the pressure change shown can be large compared to the mass change, and the tyre is neither rigid nor completely free to deform. We know this because an uninflated tyre collapses under the weight of (1/4 of) a car, but not under the weight of a wheel.
The aim (I propose) is that the pressure should be adjusted to maintain something close to the desired contact area on the road, taking into account tyre deformation as a function of load. As the tyre pressure specs are a feature of the car not the tyres, these will have to be typical tyre deformations. The effect of the cushioning from the tyre will require a change in the same direction (i.e. to support a heavier mass, use a stiffer spring); this is a not-insigificant part of the suspension of a car.
The deformation won't suddenly increase as more load is added (your last paragraph), but there may become a point at which it is worth topping up the tyres.
I'm tempted to experiment (on a bike rather than a car, where I can change the loading significantly by sitting on it).
Edit it looks like your answer from Red Act (+1) now goes into more detail on this, but I'll leave mine here as it approaches from a different perspective.
Edit2:* I tried it with a bike tyre which I was pumping up anyway. Basically the stiction in a normal pressure gauge prevents a result from being observed in a realistic situation, and I didn't have the kit to attach a U-tube manometer to a schrader (car style) tyre valve pressing in the centre pin. By running the tyre at about 1 bar and leaning a fair proportion of my weight on it I could get the needle to twitch by around its own width - detectable but not really measurable.