The main attraction of the hodograph equations is their linearity. The linearity comes with a price, however, as there are several difficult drawbacks to this method. In most cases, I don't believe the price of the hodograph transformation is worth the linearity we get from it.
Although Equations 14 and 21 are linear, they do not have constant coefficients, and the coefficients are not particularly simple. Even a complex linear equation is better than a non-linear equation; however, I don't believe the benefits of linearity overcome the drawbacks of the hodograph transformation.
I would say this transformation is most useful for inverse design of high-speed subsonic airfoils. It fits this situation well due to the combination of linear equations and a method suited to inverse design. In cases where the potential equations are acceptable, the hodograph transformation can be a simplification.