> I'm surprised the infrastructure between Stadia and regular PC was so different that Stadia was effectively its own console. I assumed it was just a somewhat thicker version of something like Nvidia's GeForce,

And Nvidia is being stifled by that. Their UX is bad and leaky (you see Windows components and errors, and you have to login into launchers, with MFA, every time when starting a new session), and they probably pay a fortune for Microsoft licenses. Meanwhile Google had full control of the stack and could do whatever they wanted with it, including optimise it to hell for latency.

An issue that Stadia had with cross play is that controller and KB/M were both very common ways to play. Controller players would not want to play against PC users with keyboards. And Keyboard users would have an unfair advantage against console players with controllers. I think one game changed which you were matchmaked against depending on which input method you were using.