The Steam Machine and Steam Frame are probably delayed, but they still intend to release them in the first half of this year.
They said they planned to have a concrete launch window and pricing by now, but hardware pricing/shortages have made that hard to pin down.
But they answered the question. Valve can only guarantee that games are compatible with Steam Overlay compatible games. They don’t want to commit to a “yes” or “no” for stuff outside Steam, and then be liable for that answer if it does not work as advertised and then get sued to the ground. So you cannot blame them.
I don’t know on Windows, but on Linux you most likely can use the controller to play on any game. It’s just not supported and guaranteed by Valve.