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.
Well any game or launcher or emulator have to support the controller or type of controller. In example Valve cannot guarantee that all functionality (like the trackpads and motion controls) are supported elsewhere.
They don’t really have to support the controller, though. Steam input lets you map controller inputs to kb/m inputs, so no degree of controller support is required. If there are any programmes that don’t work (which is possible, there are weird quirks in any system), I’ve certainly never encountered them.
Does steam input run on the controller itself or the device it’s connected to though?
It runs as part of steam, so you would need to add the program you want to use to steam as an external program and start it through steam. Steam Input then runs as a translation layer around the other program. At least that’s how it works on the Steamdeck.