Corals grow fine under white lighting. The thing that matters is the spectrum the bulb puts out. If you look at the spectra for the bulb you will see it contains different frequencies which the coral needs. The move to the bluer end of the spectrum is to highlight coral colours as they look more washed out in whiter daylight bulbs bit look at the spectra of a typical 10k bulb and you will see it still contains plenty of the blue spectrum in it.
White is relative. Daylight is 6500kelvin. basically. Most whites you see in tank are 8-16,000 kelvin.
At 16k you can grow corals just fine , at 56k too in fact. Because there’s enough blue in it. At 4,000 and less you lose almost all the blue.
Add in that you can mix a color temp in a variety of ways.
My tank looks like this in person , with led and I’m pretty sure it’s more than 20k. (I have a color meter and a a 20k radium metal halide to test and compare)