What are your viewport settings? OpenGL Core (compatibility mode)? Actually, VP 2.0 is very sensitive to videocard and drivers. In 2016 I had enormous amount of selection problems of such kind. Fortunately, in 2016.5 they're gone - at least in my case.
There might be some confusion here with how this UI feature works. If under Preferences > Settings you have the Select faces with option set to Center, face center dots will always display when in face mode (you can't disable them), and you will always need to have your cursor near the center for the pre-selection to…