Yes and no; if they're doing it right, they're using the DMA or PLL engine which runs independent of the main CPU/ALU. See https://github.com/F5OEO/PiFmRds "This version modulates the PLL instead of the clock divider for superior signal purity. The harmonics are unaffected, so the legal warning still applies."

Filtering this stuff isn't hard, but I'm realizing there aren't any good stone-simple tutorials for it. Might have to write one.

I've done a hobby project where I'm using pure RPi GPIO to drive a servo motor with PWM, so processor interrupts would ruin my day. So I can tell you that the rate at which modern Linux interrupts a running application is massively overstated. If your system is configured right, that happens maybe two or three times a second at most.

If you're running on a dedicated core, as you should, 99% of the time your application just happily runs along.

Can someone explain to me why generating a square wave creates lots of harmonics, as opposed to other kinds of waves?

The spectrum of a signal is its Fourier transform. Only sine and cosine signals have a "pure" signal with a single frequency. Anything else decomposes into a whole host of spectral contributions. Square waves are especially bad because they require contributions from infinitely many harmonics to produce the steep flanks. That is also why a clean square wave cannot exist: some kind of band limit is always present, resulting in ringing around the flanks.

See also Fourier Analysis on the Wikipedia article: https://en.m.wikipedia.org/wiki/Square_wave

Because you try to generate a square wave by adding toghether "round" waves (sinusoidal). In order to get into the "corners" you must add "round" waves with smaller and smaller diameters (period). Small periods means high frequencies. The more waves you add, the squarer the sum becomes.

A square wave IS the base frequency plus ALL[1] the odd harmonics (https://en.wikipedia.org/wiki/Square_wave#Fourier_analysis explains it better than I could).

[1]: Yes, so in reality you neither get an infinite set of harmonics, nor a perfect square wave.