The points really do converge to a circle

Just to start, no they don't. The prescribed sequence of folds has perimeter of 4 after n folds and 4 after n+1 folds. It will have a perimeter of 4 after an infinite number of folds.

In reality there are a lot of ways of "explaining" this particular result, depending on taste and the point you're trying to make. I'm choosing the tact that it's ultimately the difference between an infinite sum and an integral, which relates to the difference between the infinity of the integers versus the infinity of the reals.

"Under the hood" what we basically have is an "approximation-scheme/function" g(x) which touches another curve f(x) at points x_n such that g(x_n) = f(x_n). Although g(x_n) = f(x_n), the arc length, or perimeter, isn't determined by the values of the function at each point x_n, but rather the slopes/derivatives at each of these points and it need not be true that dg(x_n)/dx ~ df(x_n)/dx, even if g(x_n) = f(x_n).

Thus, for an x_n defined discretely/ i.e. with integers, f(x_n) = g(x_n) for all x_n DOES NOT imply that df(x_n)/dx = dg(x_n)/dx

However, in the continuum of the reals then the same statement: that f(x)=g(x) for any x DOES imply that df(x)/dx = dg(x)/dx

I think it's a perfectly valid "take" on the situation.