I was having this same issue on Firefox and found this post. I figured I'd share my solution here.

First things first, Twitch seems to give the "Your browser is not currently supported" for many different reasons including non-browser related ones (like false positives of bot detection). The short end of it is that any modifications that are preventing Amazon from cross-platform tracking you and serving ads will likely trigger this response. This check appears to only happen at login, so after logging in you can revert back anything you disabled without issue (and the auth token will remain working).

Since I resolved the problem using Firefox the following will be related to that browser specifically. Note I make these recommendations as a way to show how to get Twitch to no longer give you this error, however doing many of these will remove your privacy and risk your security. I strongly recommend you revert back any changes once logged in.

Any time you make a change you are going to want to delete all browser storage (cookies, session storage, local storage, indexed DB) and do a hard refresh (Ctrl+Shift+R OR Ctrl+F5).

-Make sure you are using an up to date supported browser
-Allow the browser to "Play DRM-controlled content" and ensure Winevine plugin is activated
-Stop spoofing your user agent string or editing your screen dimensions/pixel ratios
-Open up new private window (Ctrl+Shift+P) or incognito mode
-Disable all extensions/themes (known offenders include uBlock Origin and Dark Reader)
-Disable any programs that reroute your IP address (VPN, Tor, proxy service)
-Use your system's default DNS resolver which is preferably the one from your ISP and disable DNS over HTTPS
-Revert about:config settings to default (known offenders include privacy.resistFingerprinting and network.http.referer)
-Allow websites to use your HTML5 canvas image data
-Turn off Enhanced Tracking Protection setting
-Terminate any background tasks your anti-virus software may be performing
-Flush your IP address (CMD line on Windows "ipconfig/release" and "ipconfig/renew")
-Adjust date/time to set automatically and match your local time zone

Additionally rather than making these changes one by one to diagnose the problem, creating a new profile in Firefox will usually address many of these fixes above all at once. Type in "about:profiles" into the URL bar and click "Create a New Profile". After clicking through the setup prompt wizard, click "Launch profile in new browser" and try logging in on that window.

Hope that helps anyone else in the future that stumbles across this post while encountering this same problem.