Probably wouldn’t have been feasible - I heard developers had to compile their games with Stadia support. Maybe it was an entirely different platform, with its own alternative to DirectX, or maybe had some kind of lightweight emulation (such as Proton) but I remember vaguely the few games I played had custom stadia key bindings (with stadia symbols). They would display like that within the game. So definitely some customization did happen.
This is unlike the model that PlayStation, Xbox and even Nvidia are following - I don’t know about Amazon Luna.
Stadia games were just run on Linux with Vulkan + some extra Stadia APIs for their custom swapchain and other bits and pieces. Stadia games were basically just Linux builds.
As I understand it, GeForce Now actually does require changes to the game to run in the standard and until recently only option of "Ready To Play". This is the supposed reason that new updates to games sometimes take time to get released on the service, since either the developers themselves or Nvidia needs to modify it to work correctly on the service. I have no idea if this is true, but it makes sense to me.
They recently added "Install to Play" where you can install games from Steam that aren't modified for the service. They charge for storage for this though.
Sadly, there's still tons of games unavaiable because publishers need to opt in and many don't.
They did have a dev console based on a Lenovo workstation, as well as off-menu AMD V340L 2x8GB GPUs, both later leaked into Internet auctions. So some hardware and software customizations had definitely happened.
This is unlike the model that PlayStation, Xbox and even Nvidia are following - I don’t know about Amazon Luna.