There's also the intuition that the circumference of the circle must be less than the perimeter of the square, so if the perimeter of the polygon isn't decreasing as it gets closer to the circle, it doesn't approximate it better than the square itself.

I.e., the perimeter doesn't approach the circumference in value because it doesn't change.

It's an interesting thing to think through though, and maybe a good point about how arguments can seem intuitive at first but be wrong. On the other hand, I'm not sure that's any more true of visual proofs than other proofs.