Since I still have an HD 5770 sitting in the closet and have some downtime, I grabbed it out of the box and put it in my old SFF PC that I use with my CRT TV since that most convenient and is also of almost near-similar vintage (and is at least as close as I have because I never bought anything on 1156).
It has Windows 10 Pro 22H2 on it though. But, the HD 5770 does work with the latest patch, at least on this system.
This is the driver used; it’s the latest beta drivers from AMD for Windows 10 for the HD 5xxx series (Crimson Edition 16.2.1 Beta):
Here’s some images of it running (note that the TV is a 4:3 CRT HDTV).
It defaulted to DX11, and I didn’t check if DX11 Legacy works still or not on it.