DSR is a form os super sampling, the game is rendered at a resolution higher than the one displayed on the screen, this eliminates all jaggies and gives a great IQ.
As far as I know there is 1 console game that does this, this is one of the Lego games on the PS4 (I can't recall which one) it renders at 1920*1200 and downscale it to 1920*1080, this makes a super sharp image!
Now, going all the way to 4K then scaling back to 1080p is for IQ freaks and this should not be expected on any currently shipping console (or even 98% of PCs out there).... and while it gives you some extra information in the final image, this is by no means a replacement for 4k as you could render the image at "8k" then downscale it all the way to 4K and get an even better image (I would probably not waste ressources on this, but someone will do this one day, for sure).
Now early AA methods were simple super sampling, you would just select a multiplication factor in your video card drivers (it was never an option in the games graphics settings back then) then the image would be rendered at 2, 4, 6 or even 8 times the resolution you picked inside the game! the IQ was always great... but you ran into performance problems really fast.