What you desire is possible in Blender, for example, but it requires a dedicated Face Select mode, with a dot in the middle of every face so you can pick them through the X-ray:
That’s quite a convoluted method, and it gets visually busy very quickly as the number of faces rise, not very SketchUp-like, so I doubt something like it would get implemented in the official package.
Curic has a plugin called “Go to Level” that can pick elements under the pointer even if they’re nested within groups, but even that can only select the first face:
I’m not sure if that was a design decision on Curic’s part, or if SketchUp’s API doesn’t allow more than that, perhaps due to a potential performance hit calculating all faces that might lie under the pointer would bring.
For selecting a hidden face like face A without moving your camera, I recommend trying the Trowel tool from Curic DIO2. It’s quite straightforward: hover over an edge and hold the Shift key when you need to select a face for pushing, but it’s obscured. This will select the obscured face for you. Additionally, the Trowel tool features an ‘Isolate Object’ function that quickly isolates objects, allowing you to select faces that are otherwise not visible.
Give it a try; I believe it will greatly simplify your workflow.
To help achieve your goals, some time spent at the SketchUp Campus and at the SketchUp - YouTube channel will be very worthwhile. Both sites are from the SketchUp team. On the YouTube channel, pay attention to the Square One Series and Level UP series. They cover the basics for each tool.