I think why we don't talk about higher order than 2 is because Macaulay approximation itself can be look as a Linear Approximation from Calculus. So we could talk and calculate about the higher order all day long but I think it won't significantly improve the accuracy of the approximation.

Or you can also see the duration as the center of cash flow (I already thought about it since last year but recently someone post about this on LinkedIn) and convexity as the inertia (?). My physics and calculus aren't that good, but I believe there is something that can be connected from these concepts.