Another month down according to Statcounter at least, Linux on the desktop is doing better than ever. Take it with your usual little pinch of salt like any survey sampling though.
Had your same exact issue, and after jumping through hand-made solutions and countless clients i finally found a client that works perfectly out of the box for screen sharing with audio, has no other issues and comes with the big plus of having customization plugins Vesktop (i think its on flathub too so if your distro ships that probably get it from there).
Had the same issue here too and yes, while my “main” game got recently proton verified and i could finally get totally rid of windows, there are some few (BattleEye mostly) games with no anticheat support.
Thanks for recommending Vesktop! It works better than my previous alternative. Still doesn’t seem to be able to really do 60fps streams though…