As I understand it, the Stadia hardware is pretty standard CPUs, but GPUs without output logic/hardware. The game's screen images are output to memory, and compressed to a stream to be sent over the Internet to the Stadia Client, which decompresses and displays the images.
It's all handled through Vulkan. Instead of display they select some other type of swapchain, that's used for video stream. It should be doable to make it configurable and output the same thing for normal display.
