It's not so much the oxygen, than the moisture in ambient air.
Air is ~80% nitrogen to begin with. Most nitrogen fills that purify out of air only gets to ~95% nitrogen. And unless whoever does the filling does a minimum of 3 purge-fill cycles, they won't be up to the 95% nitrogen level.
Gasses follow the ideal gas law, whether it's oxygen or nitrogen or helium or whatever. They all expand or contract the same amount with changes of temperature. i.e. the coefficient of expansion with temperature is constant for all gasses. So less variations in pressure over temperature swings is bogus.
However, if you have moisture, as the temperature drops, or you compress the air in the tires and the moisture condenses, you have a more significant drop in pressure.
Here's the math- 18 grams of water takes up 22.4 L when it's a gas... when it condenses, it takes up 18 mL. That's over 1200 times volume contraction. So if you have 3% moisture in the air at 30 psi, if it condenses, it'll drop close to 1 psi immediately. And it's that moisture combined with oxygen that contributes to corrosion.
But then corrosion isn't that much to worry about- the outside of your tire is subjected to way harsher conditions than the inside will ever be.
The final thing, then, is that oxygen is smaller, and leaks faster than nitrogen. By how much? about 4 times faster... So purer nitrogen in tires do leak "slower".
But then, as the oxygen leaks and you put in more air, you're essentially upgrading the amount of nitrogen within the tires.
So if you can get nitrogen inflation for free, go for it. If you have to pay for it, know that it's generally only 95% nitrogen. The more important thing is that you get DRY air or DRY nitrogen.