Automatic correction of various lens distortions is possible through specialized software that takes a few camera parameters and produces an output image free of distortion.
Why don't most cameras do this? In particular, the GoPro camera suffers from serious fisheye distortion:
There are several problems transforming the curvilinear perspective of the wide angle GoPro lens to a rectilinear one.
The transformation stretches the image in certain areas and compresses it in others. This lowers the image quality as new pixels are created as approximations and interpolations from the information in the original image.
There is also a problem with the rectilinear image from a very wide lens. To make sure that lines are straight certain sacrifices must be made. The closer an object is to the edge of the frame, the more stretched it will appear. This may create unwanted forced perspective effects.
These are a couple of reasons to why the GoPro won't automatically correct for the distortions that the wide angle lens creates. Also if you want to correct for them in post, there is nothing that stops you from it.
Simply put, the distortion isn't so much a result of the lens as it is the angle of view. Since you are seeing things that are almost directly out to the side, you are seeing them from the side rather than the front.
You can apply a rectilinear correction, but doing so will result in forcing the perspective to not match up with what you are actually seeing as you will appear to be looking at the front of something by perspective, but will actually be looking at it from the side. This can produce a rather jarring image as well.
