This paper addresses the problem of tracking the position of a cable-suspended payload carried by an unmanned aerial vehicle, with a focus on real-world deployment and minimal hardware requirements. In contrast to many existing approaches that rely on motion-capture systems, additional onboard cameras, or instrumented payloads, we propose a framework that uses only standard onboard sensors--specifically, real-time kinematic global navigation satellite system measurements and data from the onboard inertial measurement unit--to estimate and control the payload's position. The system models the full coupled dynamics of the aerial vehicle and payload, and integrates a linear Kalman filter for state estimation, a model predictive contouring control planner, and an incremental model predictive controller. The control architecture is designed to remain effective despite sensing limitations and estimation uncertainty. Extensive simulations demonstrate that the proposed system achieves performance comparable to control based on ground-truth measurements, with only minor degradation (< 6%). The system also shows strong robustness to variations in payload parameters. Field experiments further validate the framework, confirming its practical applicability and reliable performance in outdoor environments using only off-the-shelf aerial vehicle hardware.