Civilian airplanes aren't using AM because of the Doppler effect. You're not accelerating that rapidly to make the Doppler effect that pronounced on the kind of radio being used in airplanes to the point they wouldn't be useful. Even if you're going hundreds of miles an hour the shift is going to be a few dozen Hz in drift. A cheap FM discriminator will be able to handle that without any problem. Doppler shift starts to matter when dealing with satellites, but not airplanes unless you're taking a SR-71 on a civilian stroll.
Doing the math, if you're going 200mph away from a station transmitting at say 121MHz, the drift frequency would be ~36Hz. Not going to be a problem.
And even then, your AM transmission still gets affected by Doppler shift as well.
Airplanes use AM because when two SSB transmissions happen at the same time you can actually hear both at the same time. If you're using FM it's either an incoherent mess or one transmitter drowns out the other.