LED PWM frequencies are FAR higher than the old CRT refresh rates.
That's not the problem here. The flicker that is being observed is not the refresh rate of the LEDs but the refresh rate of the LEDs when masked by the refresh rate of the LCD.
Try looking at a video recording of a CRT screen. The refresh rate of the CRT is plenty fast but, when sampled at the sampleing rate of the video recorder, visual artifacts are introduced. This is why newsrooms always synchronize the CRTs visible from the camera from a single source.
What the original poster described is likely an artifact of the LED and LCD hardware not being designed to work together. The manufacturer probably took off-the-shelf parts to design their product without worrying about how they interfaced with each other. Who knows, the design might have originally worked great but then someone changed one of the parts to save some money. As a result, %5 of the resulting products end up showing flicker.
This is not a hard problem to solve and I assume most high quality manufacturers have done so. But manufacturers that bundle the lowest cost components together could very well see this problem.