Who cares whether it is quackery or not: these tools have a self-evidently useful purpose which is to manage the white balance of a display as the ambient light shifts between daylight and artificial light. Whether it has any effect on sleep hygiene is entirely orthogonal.

A better analogy would be artisanal water: the touted benefits of their particular magical bottle of H₂0 may be quackery, but the benefits of hydration are irrefutable.

Purported sleep benefits aside, the reduced eye strain from warmer colors is invaluable to me. Sometimes I'm working on a ship's bridge where it's dark or under low reddish light. Turning on a bright, blue laptop screen is brutal. At home it's much more pleasant to look at than the colder daylight colors.

I don't know whether or not it helps me fall asleep, but I've been using f.lux or redshift for years now and it completely stopped my headaches from looking at the monitor at night.

Anecdotal, yes, but it worked for me.

All I know is turning the brightness control on my iPad all the way down to far left with lights out and in bed then just browsing boring stuff like news, Car and Driver, etc and it works like a sleeping pill for me. In about 15-20m of this and I can barely keep my eyes open.

Might be this minimized blue light effect as they suggest, or could just be a sort of recognized pattern/habit my brain now associates with bed. I don't know why, but it works for me. I'm getting sleepy now just thinking of it..

So? Most of the time we fail to sleep because we feel anxiety, and our brains tells itself it's never going to sleep

it's mostly mental, not physical. Hence a placebo is precisely what we need.