Over on his YouTube channel Tall Paul Tech has uploaded a video that demonstrates the FM (frequency modulation) capture effect. Apart from the costs and difficult logistics to change from AM to FM worldwide, the FM capture effect may be one additional reason as to why aircraft still choose to use AM modulation for communications instead of FM.

The FM capture effect is a phenomenon that occurs when two FM transmitters transmit on the same frequency at the same time. What will happen with FM is that the stronger of the two transmissions will be the only one heard, with the weaker one totally muted. This is in contrast to AM where both signals can be heard, albeit garbled like two people talking at the same time.

With aircraft this is important as for example if some aircraft accidentally leaves a blank transmission open, another aircraft can still transmit on top of the blank transmission and still be heard. Or allowing air traffic control to hear if multiple aircraft are trying to transmit at once, and handle communications appropriately based on urgency. The disadvantage is that without the capture effect, AM is more prone to interference from interference and atmospheric noise like lightning.  

In his demonstration Paul uses two HackRF's with their clocks linked and an RTL-SDR to simulate two transmitters and a receiver.