The issue comes down to how the device interacts with your computer. Just like the original version, the new Steam Controller has no native Windows drivers. This means the hardware relies entirely on the Steam app to function properly. If you do not have the game running via Valve’s storefront app, your shiny new gamepad turns into a useless piece of plastic.
Gamers Nexus also reported this and there are a lot of other news outlets also covering this. It is kinda the same as with the steam deck where I noticed that the inputs just don’t work until steam is launched. I really don’t like that I have to have Steam running for this controller to work. I don’t know if it is a dealbreaker for me, yet, but it really put a damper on my enthusiasm about it



There is sadly no workaround for in-home streaming. In my case, I connect controllers to my Android TV, which is easy because they (Xbox, PS, Nintendo) all use basic Bluetooth profiles. Then, I use those controllers to open the Moonlight or Steam Link app, at which point they’re forwarded on.
This wasn’t an issue when the Steam Link was a physical piece of hardware made to work with their controller, but that’s been discontinued.
I don’t understand what problem you are referring to, all that already works on the old steam controller. It has Bluetooth mode
Steam link works with the original Steam Controller through Bluetooth, I’d be surprised if the new one didn’t.