I am sorry to be blunt, but that really is cobblers. Tyres on wheels in motion flex, the flexure generates internal friction, the temperature of the tyre rises and some of the heat generated is transferred to the inflating fluid, whatever it is. Similarly, rolling friction between tyre and road generates heat and some of this is transferred to the inflating fluid. After running for some time some level of temperature equilibrium will be reached, where the heat generated is balanced by the heat radiated to atmosphere. This temperature is always higher than the temperature at rest. It is simply not possible to run an inflated tyre from cold without a rise in its temperature. This is one of the reasons why racing tyres are pre-heated - it minimises the change in working temperature when the tyre is run.
Every gas, including nitrogen, obeys the standard gas laws.
The general gas law says that pressure times volume divided by absolute temperature is a constant. So if the temperature of the inflating fluid rises, so will its pressure. There may be some small expansion of the contained volume of a tyre because of the increase in pressure, but it will not be enough to counteract the whole of the pressure rise.
Nitrogen may perform better than wet air because the evaporation of the water content of wet air will give rise to greater pressure changes for a given rise in temperature. If however, clean/dry air is used, it is a perfect gas and obeys exactly the same gas laws as nitrogen. In normal road-going vehicles the difference is so marginal as to be not worthwhile. In racing, the difference is worthwhile and it is probably easier to provide a supply of nitrogen than a supply of clean dry air.
The essential reason why nitrogen is used in certain applications such as aircraft tyres and hydraulic/pneumatic systems is safety. Since it contains no oxygen, nitrogen will not support combustion.
I am sure we have been through all this before . . . . .