The fast-growing demand for fully autonomous robots in shared spaces calls for the development of trustworthy agents that can safely and seamlessly navigate in crowded environments. Recent models for motion prediction show promise in characterizing social interactions in such environments. Still, adapting them for navigation is challenging as they often suffer from generalization failures. Prompted by this, we propose Social Robot Tree Search (SoRTS), an algorithm for safe robot navigation in social domains. SoRTS aims to augment existing socially aware motion prediction models for long-horizon navigation using Monte Carlo Tree Search. We use social navigation in general aviation as a case study to evaluate our approach and further the research in full-scale aerial autonomy. In doing so, we introduce XPlaneROS, a high-fidelity aerial simulator that enables human-robot interaction. We use XPlaneROS to conduct a first-of-its-kind user study where 26 FAA-certified pilots interact with a human pilot, our algorithm, and its ablation. Our results, supported by statistical evidence, show that SoRTS exhibits a comparable performance to competent human pilots, significantly outperforming its ablation. Finally, we complement these results with a broad set of self-play experiments to showcase our algorithm's performance in scenarios with increasing complexity.