Time-optimal model predictive control is important for achieving fast racing drones but is computationally intensive and thereby rarely used onboard small quadcopters with limited computational resources. In this work, we simplify the optimal control problem (OCP) of the position loop for several maneuvers by exploiting the fact that the solution resembles a so-called ‘bang-bang’ in the critical direction, where only the switching time needs to be found. The noncritical direction uses a ‘minimum effort’ approach. The control parameters are obtained through bisection search schemes on an analytical path prediction model. The approach is compared with a classical PID controller and theoretical time-optimal trajectories in simulations. We explain the effects of the OCP simplifications and introduce a method of mitigating one of these effects. Finally, we have implemented the ‘bang-bang’ controller as a model predictive controller (MPC) onboard a Parrot Bebop and performed indoor flights to compare the controller’s performance to a PID controller. We show that the light novel controller outperforms the PID controller in waypoint-to-waypoint flight while requiring only minimal knowledge of the quadcopter’s dynamics.