You are misunderstanding the compression. The face appears flat because you are observing it from a distance and thus the angles of it ARE flat. If you move in closer, you are viewing it more from the side/off-axis and you get more depth. The focal length is simply used to crop the image for you.
You could take a portrait with an 135mm lens at 3 meters or you could take a photo with a 14mm lens and crop it down to the same field of view and the flattening would be the same. It would have an impact on other aspects of the photo, but not the angle at which the face is being seen or the compression you describe.
Similarly, the field of view of the lens is a direct result of the focal length of the lens and the size of the image circle it projects on to the sensor. You could theoretically make a lens that would project more information for that focal length (through a bigger image circle), but would then need a different size sensor to capture the expanded image circle. This would be the same as having a <1 crop factor.