Not on your screen, but what do you see in your periphery? You see your hands on the HOTAS, or keyboard and mouse, if you're into such a things. You see your chest. You see your nose. You see your cheeks. You see a vague outline of your eye's FoV. You see your legs. You see your feet.
Artistic impression of OP

OP - a lot of what you are talking about is actually your brain filling in details that you actually are not directly seeing.
Maybe FD could do more with FoV sliders for people who want a wider FoV, although playing with them to extreme levels would possibly make things look strange. I know when you use the existing slider and put it all the way over, things start to look a little stretched.
Sounds like what you actually need is a VR headset for that full immersion.