The physics behind sidewall flex in an aircraft tire and sidewall flex in a car tire are the same.
The primary reason for blow-outs in car tires is underinflation, and it doesn't take much to cause significant heat rise and eventual tire failure. It also doesn't take a significant value of underinflation to cause excess shoulder wear, contributing to rapid tire wear and degeneration, which also assists a car tire in failing.
On a short trip with a of braking, of course the majority of the heat received by the wheel assembly will be through braking. However, underinflation will cause temperature rise in the wheel regardless of braking. A long road trip with no brake application on the highway can still cause a tire failure due to the weakening of the tire, and the temperature rise from sidewall flex, alone.
For normal applications, nitrogen isn't available for car tires and a rule, and isn't absolutely necessary. It's still preferred, where available.
Pressurized atmospheric air in a tie sustains combustion, whereas nitrogen does not. Blowouts in tires with a much more flammable mix, using air (and worse, using propellants from inflation cans, and products from inflation cans) can contribute to catastrophic failures and explosions, as can hydrocarbon products from the tire itself. Nitrogen discourages combustion. It can mean the difference between a tire that fails explosively, and one that simply goes flat. One is more likely to see this difference manifest in Phoenix in the summer, vs. Minnesota in the winter, but it's also another reason that performance vehicles do use nitrogen in the tires.
So yes, sidewall flex is a consideration, and is not a myth.