Yeah this is important when you print images, since printers can reveal banding artifacts you can’t see on a monitor. I learned this the expensive way printing a batch of large format images that had gradients in them, without dithering. This is also how I learned one way that Photoshop was for professionals and GIMP & ImageMagick weren’t; At the time (and this was years ago, to be fair), Photoshop dithered by default when going from 16 to 8 bits, while those popular open source tools did not and also did not even have good ways to do it manually by adding the right amount of noise before changing the bit depth. I don’t know what the open source tools do today, maybe they’ve fixed it, I haven’t checked.

The counterpoint is that you should just keep your images at 10 bit minimum or 12+ bit until the OS/monitor can dither for you. Lossy compression will do much better with a 10/12 bit image than a dithered 8 bit one.

I'm only vaguely familiar with dithering/half-toning. What does "component" refer to here?