This article investigates the tracking control problem for Euler-Lagrange (EL) systems subject to output constraints and extreme actuation/propulsion failures. The goal here is to design a neural network (NN)-based controller capable of guaranteeing satisfactory tracking control performance even if some of the actuators completely fail to work. This is achieved by introducing a novel fault function and rate function such that, with which the original tracking control problem is converted into a stabilization one. It is shown that the tracking error is ensured to converge to a pre-specified compact set within a given finite time and the decay rate of the tracking error can be user-designed in advance. The extreme actuation faults and the standby actuator handover time delay are explicitly addressed, and the closed signals are ensured to be globally uniformly ultimately bounded. The effectiveness of the proposed method has been confirmed through both theoretical analysis and numerical simulation.