There are two ways to dim LEDs: linear regulation and some sort of pulse modulation. Linear regulation is wasteful and you're pretty unlikely to encounter it, especially in battery-powered devices such as phones or laptops. Pulse modulation is common.
Human vision has a pretty limited response speed, so it seems pretty unlikely that PWM at a reasonable speed (hundreds of hertz to tens of kilohertz) can be directly perceived. That said, it can produce a stroboscopic effect, which makes motion look weird and may be disorienting in some situations. So I don't have a problem believing that it can cause headaches in predisposed individuals.
You can dim your laptop screen in a darkened room and wave your hand in front. Chances are, you're gonna see some ghost images.
Other than adjusting the frequency, pulse modulation can be "smoothed" in a couple of ways. White LEDs that contain phosphor will have an afterglow effect. Adding capacitors or inductors can help too, although it increases the overall cost. But that doesn't make the display "PWM-free", it just makes it flicker less.