Yes to the last bit, and in the simplest case (for luminance) just take the luminance pixels in 2x2 blocks and average the four. That will give you 1920x1080 "new" values for luma information, each independent. (Practically, I'm sure there's a better way to downscale.)What you described is some form of Luma subsampling, and I don't know if that is something people do, because you want each pixel to have its own independent luma information.
If they don't, more fool them!This is more of a theoretical thing you could do, but no one actually does.
OK, more seriously, everyone has to work within the software they have and often exactly how these behave can be hidden. The real point is - it is theoretically quite possible to get "true" 1080 4:4:4 from 4k 4:2:0. Same obviously for 1080 4:2:2. Whether or not any given software is capable of doing it is another thing entirely. But there's no problem with the theory.