Late response but for anyone with an issue like this, try making the 2nd display your default by going to System Preferences > Displays and dragging the menu bar to the display you want to play the game on.

I have the same problem. Steamlink (or game streaming) ALWAYS starts on the big tv connected to the macbook pro. I also tried putting the macbook as main screen, but steamlink / streaming starts on the big screen. You can't drag the desktop to the other screen. If you drag the window (if you succeed), once you click it will either turn black or crash completely.

The only way is really setting the screen you want, as your only screen. At least this is what I learned from this whole morning of testing....

I had this issue too. I started playing age of empires on a bigger display, but couldn't see my resources or population count because of the top-bar. When I tabbed out of the game (via swiping on the touchpad) to google for this issue and tabbed back-in, the top bar was gone, all of a sudden. Maybe this helps anyone.

With time I learned a trick that I am not sure I can explain here. Basically you start the streaming and you see on which screen it starts. You go to the top edge of the screen to call up the different desktops and full screen apps, and you drag the steam stream window to the screen that you actually want to use. I do not remember exactly how to trigger this but it was my solution for me.

If you hit Ctrl+UpArrow to bring up mission control and mouse up to the top where it says Desktop 1 etc. they'll turn into little window previews and you can drag the Steam Link one from one monitor to the other

Yeah that is the only way I found to work ALMOST consistently. Sometimes the layout of the steam link app gets weird but once the game starts it's fine.