The Chaikin curve is a quadratic B-spline, which is an approximating spline rather than an interpolating spline. Chaikin is very useful for lots of applications, but the reason it doesn’t fit the stated constraints in the article is because it doesn’t interpolate the control points, and doesn’t “naturally” stop at the endpoints of the curve either (though it’s obviously very easy to make your Chaikin curve hit endpoints if you want to). Playing with interpolating quadratic splines is a fun exercise!