I recommend not logging out, as you'd then be affected by this.
I've opened a brand new Firefox instance and got "Your browser is not currently supported. Please use a recommended browser or learn more here." (linking to https://help.twitch.tv/s/article/supported-browsers?language...) on the login screen.
The login made a zero-payload POST to https://passport.twitch.tv/integrity and it responded with 400 and a JSON body {"error_code": 5025, "error_description": "integrity failed", "error": "Oops! We encountered an unexpected error. Please try again.", ...}.
It seems that this is not about GNU/Linux, though, as it happens at random (searches for `twitch "integrity failed"` produce results from all sort of platforms and browsers). Must be that some pointy haired boss had some important ideas about security.
I was able to log in from a Firefox on a different GNU/Linux system, so it's not like those are always blocked. I suspect there's some User-Agent whitelist or similar kind of nonsense (but looking at the console logs and bunch of WebGL errors it certainly tries to fingerprint the system), but I'm too lazy to investigate this any further.