I think you are right, in the case of the trebuchet, it just computes a black box approximation of the inverse of the system. The difference being that an analytic inversion would solve the problem for all wind and target conditions, while the NN solution will only work for for value ranges used to obtain the data that trained the NN.
In the case of the inverted pendulum, again the disadvantage of using NN is the black-bock nature of the control algorithm. As a control engineer this gives me the chills, because using black-box algorithms tell you nothing about the robustness of the closed loop system. With model-based control, at least we have strong mathematical tools to guarantee that the closed loop will be robust enough to handle variations outside the data that we used for training (our model). With black-box algorithms like NN you have no guarantees. I would not get into a plane controlled by a NN for sure, look what happened with the 737MAX when software engineers thought they could solve dynamical system problems.
In the case of the inverted pendulum, again the disadvantage of using NN is the black-bock nature of the control algorithm. As a control engineer this gives me the chills, because using black-box algorithms tell you nothing about the robustness of the closed loop system. With model-based control, at least we have strong mathematical tools to guarantee that the closed loop will be robust enough to handle variations outside the data that we used for training (our model). With black-box algorithms like NN you have no guarantees. I would not get into a plane controlled by a NN for sure, look what happened with the 737MAX when software engineers thought they could solve dynamical system problems.