Why is my Linear FOV GoPro Hero 4 Black video distorted at edges?
The distortion you are seeing has nothing to do with ProTune or other codec settings. Its perspective distortion caused by the wide focal length of the GoPro Lens + The low quality of the GoPro Lens itself. 

When I say low quality, I mean that in relative terms. The GoPro lens actually produces a pretty amazing image for what it is. But even when you enable the Linear FOV mode, the camera's software can only do so much to reduce the tearing - specifically in the corners of the frame, that you are seeing. It does a decent job overall, but the lens on the GoPro in Wide FOV is roughly 17mm equivalent to a full frame lens. 

If you want nearly zero distortion at that focal length... The Zeiss 15mm released a few years back is fantastic, but it's $2500 ish just for the lens itself. 

The 16mm and 14mm Arri Master Primes are practically flawless in terms of edge and corner distortion, remarkable actually even at their widest f1.4 t-stop. But for that look, you have to rent or shell out $30,000 for the lens.  

So that is basically the reason. You will only get "so good" with a cheap plastic lens on a $400 camera at that focal length. And there is only so much the in camera software can do to process that distortion out. Even doing a more detailed correction in post will get you so far. 

The problem has nothing to do with the settings. Thats as good as it gets with the GoPro, in terms of it's wide angle ability to capture a relatively flat looking image. Flat in the middle, but the closer you get to the corners the worse it's going to get. 

My recommendation is if you can't live with it, shoot from a farther distance and crop in in post. I think judging by the image had you just backed up perhaps 5 or 6 feet, you'd be able to crop in just a bit and remove "the worst of it".