There's 100% ALREADY a mechanism for this. Most companies just dont do it because it's free advertising.
If you dont want your game streamed just include it in the EULA and say if you want to stream it you gotta pay for a license.
In fact you dont even need to put it in a EULA, you have the right to tell youtube to take down Let's Plays of your game. People literally cant comprehend this happening because of how normalized it is but Let's Plays and streams are still absolutely in a legal gray area.
It's so gray in fact you can even put out a notice saying you give everyone the right to stream your game and then just turn around and strike down content anyway. Yes that situation actually happened.