Streaming game to laptop while gaming on other screen

As you said, "everything should be possible with a pc", but that's not a 1:1 of "everything is currently implemented on a PC".

The heart of the issue is that steam Remote Play doesn't run as some background process on the PC, leaving other activity to do whatever. When you run a Remote Play all it does it stream your computer screen to the remote device, and then accept inputs back from the remote device.

As others have mentioned, Windows doesn't play particularly well with trying to have multiple windows in focus at once. Generally, you can only select one window, and that is the window that will receive inputs from your devices.

There are some subtle exceptions, like if you point to a different window and scroll your mouse wheel, it will pass that through to the window you are pointing at rather than the window in focus. However, the issue is that for Keyboard and Gamepad input there is no onscreen cursor that you can point wherever, there is only the typing caret in the focused window.

TL;DR: At present there isn't a clean implementation of how to play two games at once on a single PC

If you are physically in the same building with both PCs, plug your controller into the gaming server and stream with any other software, could even use Twitch. A bluetooth controller could in theory work 30ft away, same for 2.4Ghz controllers, USB is rated for 15ft.

Personally, I ran a 50ft HDMI cable between my gaming PC and a TV/monitor because it gave me far better quality than streaming it, but I was using Steam Remote Play to start the game and for the controller input.

If distance is an issue, you would likely need a 3rd device (raspberry pi?) to handle the controller with some sort of software to handle USB over IP.