These quadrature techniques are strictly more powerful than adaptive sampling. Part of the motivation for this blog post was to raise awareness of quadrature, as it's an excellent general method when you want to numerically approximate an integral.

One reason to compute with excess accuracy is so that you can compute partial derivatives of arclength with respect to some parameter in the inner loop of an optimizer, to drive Newton or related solving. If the estimate has significant errors, then it can throw off the outer optimization loops. Of course, this only makes sense if it's cheap to compute the inner terms; if you have to pay dearly for accuracy, then use a more robust optimizer.