I think it's because no matter how many corners you cut it's still an approximation of the circumference area. There's just an infinite amount of corners that sticks out
It's a fractal problem, even if you repeat the cutting until infinite, there are still a roughness with little triangles which you must add to Pi, there are no difference between image 4 and 5, the triangles are still there, smaller but more. But it's a nice illusion.
Because you never make a circle. You just make a polygon with a perimeter of four and an infinite number of sides as the number of sides approaches infinity.
That approach works for area but not for perimeter, because cutting off the corners gives you a shape whose area is closer to the circle's, but it doesn't change the perimeter at all.