In my research thus far, the AMOLED screen technology is to blame. When the display circuits get old or damaged, they don't sustain current at low power settings. In LED screens, a back light controls brightness, but AMOLED screens send current to each pixel for brightness control. If the circuit is damaged, or any corrosion has built up inside, this could attenuate the current below the point required to sustain light flow. I see this problem with some LED and CFL lights on motion sensor switches. Just enough current goes through the motion sensor to keep it's circuits active, and that current causes a flicker in the light.
With the screen though, since the problem is only resolved when you increase the brightness just a bit above minimum, it seems anything that lowers the display power (auto brightness for example) will cause it to flicker if this setting gets too low. I also noticed that before the screen locks on timer, it dims to low power for a few seconds and flickers as a result. There are apps that attempt to resolve this by putting a translucent black screen over your desktop and apps. This keeps the power up on the pixels but adds a black overlay, kind of like a screen filter. You can adjust the darkness if you need it to save your eyes when reading in the dark but it doesn't help save power any. They also require a rooted device, which could void warranties. The reason for this I believe is because apps are supposed to operate in a way they don't intrude on each other, and a filter that lays over your screen does in fact alter the output of your display rendering. With the added security concerns of these apps possibly collecting screen shots and sending them to another site, I would avoid this. Best to use a physical screen overlay you can put on or remove as needed as long as it doesn't interfere with your touch screen options.
Ultimately, these devices were meant for replacement every 2 to 3 years so if you're like me and you like to make your investment last, best thing to do is apply any work around you can come up with. A repair shop may be able to wipe the circuit boards down with a solution that clears away corrosion, but it won't repair any weakened pixels. I'm still using a Samsung S4 and only now that I have verified it won't upgrade to the Android 6.0 OS am I considering replacing it within the next year or so.