Xbox – How to mute Kinect’s mic without leaving the game

No, I don't think so. The vast majority of games that use Kinect require it in order to play the game. I only know of one game that supports both the normal controller and Kinect - Child of Eden.

I had this issue and it turns out that my second controller constantly thought it had a headset plugged in. Thus, giving you the mic icon in the chat window, disabling your kinect for chat and no speech coming out of speakers. Try using your second controller as the first player and see if the kinect chat feature works.