They didn't promise a Linux client. Idk where you saw that, but whenever they said future clients were coming, I never saw Linux mentioned.
A couple months ago though (if that), they did say Linux support was coming in the form of a browser client, though that's already a thing, you can already use GFN on Linux w/ a chromium-based browser.
You can now run GFN in the browser on Linux (It might need Chrom(e/ium)).
The downside is that if you have a non 16:9 resolution you will have to change your desktop resolution to support fullscreen with black borders, otherwise the content renders off the bottom of the screen.
I tried it just now (with 16:9) and unfortunately mouse doesn't work in game and I see a window bar at the top (half of it actually, I can double click on it and then I see full bar + game inside this window).
I'll need to experiment with it more, I'll try to make it work in Firefox as it has better codes under linux.
The only downside I see is that there is no Linux client (or a browser one, which they promised) and I need to do a strange setup of:
1. Starting game on Android
2. Connecting bluetooth mouse there (so it won't kick me out because I'm not doing anything on the phone)
3. Starting Steam and using Steam Remote Play
(Thank you a anonymous user on reddit for letting know that Steam Remote Play works there)
(with each step cursing Google for not enabling video out on Pixel 4 USB-C)
It is a bit of PITA, but I'm glad it works (other option of using a VM was even worse).