There's a lot of great Fourier visualizations out there (3Blue1Brown has a series on it as well). A great intuition pump would be a game where the player has to recreate a waveform given to them, by toggling on/off switches for each fundamental frequency (and phase).

Has anyone ever seen something like that?

I created a small piece trying to build intuition, my approach was to think of it as probing with test signals, so instead of recreating a waveform, it's more poking at it with sines and seeing how well the test sine matches the target signal.

https://dsego.github.io/demystifying-fourier/