Autonomous 3D scanning of open-world target structures via drones remains challenging despite broad applications. Existing paradigms rely on restrictive assumptions or effortful human priors, limiting practicality, efficiency, and adaptability. Recent foundation models (FMs) offer great potential to bridge this gap. This paper investigates a critical research problem: What system architecture can effectively integrate FM knowledge for this task? We answer it with FlyCo, a principled FM-empowered perception-prediction-planning loop enabling fully autonomous, prompt-driven 3D target scanning in diverse unknown open-world environments. FlyCo directly translates low-effort human prompts (text, visual annotations) into precise adaptive scanning flights via three coordinated stages: (1) perception fuses streaming sensor data with vision-language FMs for robust target grounding and tracking; (2) prediction distills FM knowledge and combines multi-modal cues to infer the partially observed target's complete geometry; (3) planning leverages predictive foresight to generate efficient and safe paths with comprehensive target coverage. Building on this, we further design key components to boost open-world target grounding efficiency and robustness, enhance prediction quality in terms of shape accuracy, zero-shot generalization, and temporal stability, and balance long-horizon flight efficiency with real-time computability and online collision avoidance. Extensive challenging real-world and simulation experiments show FlyCo delivers precise scene understanding, high efficiency, and real-time safety, outperforming existing paradigms with lower human effort and verifying the proposed architecture's practicality. Comprehensive ablations validate each component's contribution. FlyCo also serves as a flexible, extensible blueprint, readily leveraging future FM and robotics advances. Code will be released.