I remain Sceptical
I don't understand why it would be any different. Are you sure you haven't just talked yourself into thinking it's better? A little bit of self-delusion does wonders for your confidence
If you fill a tyre with nitrogen, and an identical tyre with dry air to the same pressure, I can't think of any practical physical test you could apply to the outside of the tyre to tell which is which.
The universal gas law is called
universal for a reason. All gases behave the same way when compressed. Tyre flex is a function of tyre wall rigidity, not the inflation gas. Tyre will be the same shape. The road contact patch will be the same. The tyres will react pretty much the same way to temperature (difference in Cp netween O2 and N2 is marginal, and air is 78% N2 anyway).
I suspect that any marginal differences in performance might be more to do with the moisture content of normal air, which could either condense or evaporate (depending on temperature) and change the tyre pressure slightly, which is why I said "dry air" above.
I think the main reason it's used in aircraft and race cars is to prevent degradation of the tyre rubber from oxidation. Although I wouldn't have thought that would be an issue with race cars because the tyres don't last long enough to perish. I suspect the reason it's used in racing is because a nitrogen cylinder is a simple way to get a reliably dry gas at the trackside. Eliminating the moisture component ensures a repeatable setup.