Typical "dry air" sources available track side still has a lot of moisture in it.
For racing applications, the air in the tires are typically removed (towards but above vacuum) before nitrogen is added, to achieve better than 95% Nitrogen, same as in aviation.
Pretty severe thread drift here, but even on a really hot day with 100% humidity, you're only looking at ~1% water vapor content - so 1.6 times 1% at one bar means that my H2O vapor saturated air hot tire will increase in pressure about a tenth of a psi or so more than if I used absolutely dry nitrogen. Pretty much in the mud.
Granted, my racing was amateur, not professional, but I never, ever saw "air removed" from tires after beading prior to inflation (and I'm talking hundreds, if not thousands, of tire mountings). I did however, on a regular basis, see soapy water used as a lube to seat the bead. Some of us used a water-free compound to lube the bead to allow it to seat (the stuff I used was called "Tire Snot") but we were not in the majority (although we did tend to run up front). Most of the time, the pressure required to set the bead was higher than the running pressure so they simply bleed the tire down to something close to the desired running pressure. On occasion, the tire valve was removed and all pressure allowed to escape, then the valve installed and the tire inflated to the desired pressure.
Not only did I never see a vacuum used to evacuate the tire, I never even saw the needed equipment to do it. And again, my point has always been there is no meaningful difference between
dry air, and nitrogen when it comes to how tire pressure behaves. Racers use nitrogen because it's cheap and readily available, not because it's better than dry air.