Do I need an STM lens for face detection live video with Canon?
The AF subsystem is part of the camera. It is not affected by the lens.
Face detection is also part of the camera.

STM only affects the speed of focus for a lens (assuming that the camera feeds it with correct information.)

So yes, STM lenses are better for video when coupled with a camera that has fast focus on its own. 

For your problem there are two solutions

1)Deactivate automatic focus and select a big aperture (like 11) so that everything is in focus
2)Deactivate automatic focus and manually focus the lens on your own.

All hollywood movies, tv broadcasts, commercials etc use manual focus.