Fortunately OBS has native game capture plugins available on Linux now (not officially, you may have to install it from Github or the AUR). So you could use that to stream to YouTube or twitch for the time being.
But Live broadcasting/streaming works right from Steam without third party tools. I did that years ago: https://steamcommunity.com/?subsection=broadcasts Or was it not on Linux? Right now I can setup and broadcast and the stream appears as Live (0 viewers). And when clicking the stream in Steam client or in browser, would not start playing. But I could swear it did years ago.