Why does FM sound better than AM?

Someone gave me an analogy some time ago that made a lot of sense.

If you shine a flashlight through a tree blowing in the wind and vary the brightness to convey information, the signal can get distorted pretty easily.

However, if you have a constant brightness source and vary the color, it’s a lot easier to figure out what the source is trying to convey.

This makes a lot of sense so long as your source of noise is something like a tree swaying in the wind, ie something that interferes with the amplitude. If instead the source of noise is uhhh a piece of stained glass swaying in the wind then blinking the flashlight is the better bet. I guess it just turns out radio interference is more like the tree. But why?

Stained glass won’t (I think) shift any frequencies. It will attenuate different frequencies differently, but it won’t make up new ones.

So when the signal frequency changes, you’ll still see that change, but the light might get brighter or dimmer at the same time due to the stained glass. But you don’t care about the brightness to begin with.