I was in the same situation. I installed Linux on my old garage PC, installed the Zwift app on my phone, and then used scrcpy to mirror my phone to the PC (connected via USB cable).
If your phone is capable of running the app, this solution avoids all the complications of trying to get Zwift running in Linux and getting everything to connect. scrcpy also allows you to interact with the mirrored app using your PC mouse and keyboard.
You may find the default graphics quality is a bit lower, but you can tweak scrcpy to improve this.
This solution replaced my existing W10 setup with no problems. It also makes it very easy to deal with upgrades or try different apps. I binned Zwift after the price rise, and all I had to do was install IndieVelo on my phone and carry on mirroring.
When I tried Linux in the past, I always gave up because it was too frustrating dealing with the all the wee tweaks to get everything running properly. I used Linux Mint this time and was very impressed with how easy it was to install it and get a running system, without any fiddling. Very close to Windows ease of use.