millis and micros always returns 0 (or other constant)
millis and micros appear to use timer 0 internally. Using that timer to also generate a PWM signal messes with their ability to function properly.
The solution is to either use a different timer to generate the PWM signal, or to roll a custom equivalent of those functions.

Why does the official arduino documentation not specify that millis and micros occupy timer 0?
Why are there no Google results mentioning timer 0 when searching for "arduino millis returns 0", "arduino millis constant", "arduino millis not counting" and the like? (There are results with the same symptoms, but for different reasons.)
Who knows.