Why does FM sound better than AM?

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?

In this analogy, the AM and FM signals you receive aren't usually experiencing interference, they are experiencing multipath effects which includes things like path loss, attenuation, reflections, and so on. This is driven by geometry. You also have gaussian noise that the receiver has to deal with.

You model this by taking your signal and convolving it with the channel vector. Usually the channel vector is a finite number of dirac deltas. Each delta is a different reflection. They are like echos. They can cause the signal to constructively and desconstructively interfere with itself.

I haven't seen the math, but I am guessing this doesn't do as much to the frequency of the signal compared to the amplitude.

The stained glass would change the amplitude of some light selectively. But because the FM radio works at different distances, I wonder if it must have some way of adjusting for different amplitudes anyway?

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.

Yes, stained glass is like band filter, they let through a particular frequency range, while reducing those outside the range. Your FM receiver will still lock on the desired frequency as long as their is enough signal strength. It's kind of the same as listening to an emitter that is very far while being very close to another. Of course, it'll stop to work at some point depending on minimum signal-to-noise ratio.

a better analogy for frequency domain interference would be something like the spinning flashing lights on a fire engine or utility truck occasionally shining colored light on your detector.

In the stained glass case, maybe you need to go digital where brightness and color don't matter, but only on-off state.

You'd be surprised by the amount of brightness and color produced if you are turning things on-off sufficiently fast.

Related: lots of optical illusions.

I think this is reasonable analogy, for FM interference, and it points out why FM is resilient to noise: the flashing lights have to be relatively bright (high amplitude) to interfere with your color based scheme.