Verdict: No
No. PC Game Pass needs the Xbox app, which Microsoft only make for Windows. The cloud streaming side runs in a web browser, but Microsoft list Windows PCs and Macs rather than Linux.
Installing and playing Game Pass games on a PC goes through Microsoft's Xbox app, and the Microsoft Store listing for it offers Windows only. There is no Linux version and no way to install those games directly.
Xbox Cloud Gaming is a different and smaller thing. Instead of installing a game you stream it from Microsoft's servers to a browser such as Edge, Chrome or Safari, so nothing has to run on your own machine. It needs a fast steady connection, around 20 Mbps for a PC, and it covers a selection of games rather than everything on Game Pass. Microsoft's list of supported devices names Windows PCs and Macs, phones, tablets, consoles, some smart TVs and handhelds, and does not name Linux.
Your subscription is not wasted if you keep a Windows machine or a console. It is the same account and the same library.
If Game Pass is the main reason you play on a PC, this is one of the strongest arguments for keeping Windows on a second machine or a second drive.
Not a subscription, but Valve put real effort into Linux and a large part of the Steam catalogue runs well. You buy games rather than renting a library.
Your files: A separate store. Nothing from Game Pass carries over.
Covers Epic, GOG and Amazon libraries on Linux. Epic and Amazon give away free games regularly, which softens the loss.
Your files: Separate accounts and separate purchases.
One stubborn program doesn't have to decide the whole question. You can keep Windows on the same PC and restart into it only when you need Xbox Game Pass. Everything else you do can move.
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