Playing games from your mobile phone on your laptop with a controller?

I'm currently playing a game from my mobile phone (Galaxy S6) by projecting the screen on my laptop (I already know how to do that).
But now I want to play it with my Switch pro controller. I read that you can only use the controller with Android 10 (my phone only has Android 7 and can't have 10).

Is there another way to play my game with the Pro Controller? Can I somehow just connect the controller to the laptop? But then I can't play the game with it, can I?

No, you can't then. Your notebook is only the output device for projection. The controller should then be connected to the smartphone. Since your cell phone is apparently not suitable for this, unfortunately it doesn't work.