Trajectory planning and control have historically been separated into two modules in automated driving stacks. Trajectory planning focuses on higher-level tasks like avoiding obstacles and staying on the road surface, whereas the controller tries its best to follow an ever changing reference trajectory. We argue that this separation is (1) flawed due to the mismatch between planned trajectories and what the controller can feasibly execute, and (2) unnecessary due to the flexibility of the model predictive control (MPC) paradigm. Instead, in this paper, we present a unified MPC-based trajectory planning and control scheme that guarantees feasibility with respect to road boundaries, the static and dynamic environment, and enforces passenger comfort constraints. The scheme is evaluated rigorously in a variety of scenarios focused on proving the effectiveness of the optimal control problem (OCP) design and real-time solution methods. The prototype code will be released at https://github.com/WATonomous/control.