How does Stadia advance Linux+Vulkan? The games run in a data center (and they're not running a standard Linux port, the presentations I saw made it sound like they were running basically another console port).
More gaming developers becoming familiar with Linux+Vulkan, more games ported to Stadia's Linux, all that means more potential releases for normal desktop Linux as well. Not all who target Stadia will clearly do it, but potential is surely better than without Stadia. So I see it as net positive for Linux gaming.
Still, the games will have a vulkan renderer and run on some form of linux, making an official port to the actual OS more likely.
The Xbox One has an NT kernel and DirectX as the graphics API. Stadia's use of Linux and Vulkan are probably about as helpful for a port to desktop Linux as an Xbox One game is for a Windows port.
That analogy is the biggest thing to happen for Linux porting since Valve's Proton SDK - all future MS-funded Xbox One games do come to Windows (albeit as UWPs) and run great.
I think "MS-funded" is doing a lot more of the work there than the kernel and graphics API are.