"This website is not compatible with your device"
I can see this show up on Youtube (why not - under Google's control, and they want you to watch the ads on their official browser) and on banking apps. Initially. In the longer run, it either withers and dies, or it leads to antitrust action. I really can't see another way.
This will probably be implemented by every streaming service very quickly to try to prevent piracy (which won't work), and will only end up harming people who just want to watch on more freedom-respecting browsers or operating systems
It's already not possible to login to Twitch on Linux.
It rejects Firefox and Chrome outright. The solution is to use either browser on Wine, then copy the session cookies over.
FWIW streamlink [0] and streamlink twitch GUI [1] are FOSS solutions for watching twitch on GNU/Linux:
streamlink "https://twitch.tv/$streamer" best --twitch-disable-ads --player mpv