Unmanned aerial vehicle (UAV)-enabled positioning that uses UAVs as aerial anchor nodes has been envisioned as a promising solution for providing positioning services in harsh environments. In previous research, state sensing and control of UAVs was either ignored or assumed to be performed continuously, resulting in system instability or a waste of wireless resources. Therefore, in this paper, we propose a quality-of-service (QoS)-oriented sensing-communication-control (SCC) co-design scheme for UAV-enabled positioning systems. We first establish mathematical models of UAV state sensing and control. Then, we analyze the influence of sensing scheduling and transmission failure on the stability of UAV, as well as the performance of positioning services in the presence of UAV control error. Based on these models and analysis results, we further study the problem of minimizing the amount of data transmitted by optimizing the sensing scheduling and blocklength allocation under the condition of meeting each user's requirement on position accuracy. Finally, a heuristic algorithm is developed to solve this mixed-integer nonlinear problem. Numerical results demonstrate the validity and superiority of the proposed scheme. Compared with two benchmark schemes, our scheme reduces the failure rate or resource consumption of positioning services by more than 75% or 80%.