During the ongoing Google for Games Developer Summit 2022 Keynote, one of the Google team just did a talk on "How to write a Windows emulator for Linux from scratch" to help Stadia.
They already have some existing work available to developers who want it, including their "Stadia Porting Toolkit" which actually uses DXVK to translate Direct3D to Vulkan (since Stadia is a Linux system). However, this translator seems to be their newer approach to running Windows games on Stadia.
With only three people working on it, presenter Marcin Undak said they were able to get some games to run but it's still in the R&D phase and not production ready yet with it taking considerable time to work per-game. However, work is ongoing to hook it all up together with Wine and Proton (for 32-bit games specifically they said) which is one of the paths they will be considering going forwards.
There's already Wine and Proton, which are mentioned in the talk as well but it seems Google wanted something different. Why though? One major reason is how thin Stadia is as a platform. While it is Linux, it's stripped down to the core and doesn't have everything Wine needs to build. There's also a ton of things they don't have to care about that normal desktops do and so "90%" of Wine is simply not needed they said. Additionally, Wine is also apparently challenging when it comes to the build system and debugging.
Quite a technical talk but it's quite nicely presented, to give you a basic overview of what they're doing without going into super tech-heavy details to fry your brain if you're not a developer.
They are also finally about to open the Stadia store without the need to be signed in, so you will be able to actually look around. Incredible it took them this long to do such a basic thing, I can't imagine how many people that put off. Just think if Steam didn't let you view store pages or anything without a login.