As autonomous driving technology matures, the safety and robustness of its key components, including trajectory prediction is vital. Although real-world datasets such as Waymo Open Motion provide recorded real scenarios, the majority of the scenes appear benign, often lacking diverse safety-critical situations that are essential for developing robust models against nuanced risks. However, generating safety-critical data using simulation faces severe simulation to real gap. Using real-world environments is even less desirable due to safety risks. In this context, we propose an approach to utilize existing real-world datasets by identifying safety-relevant scenarios naively overlooked, e.g., near misses and proactive maneuvers. Our approach expands the spectrum of safety-relevance, allowing us to study trajectory prediction models under a safety-informed, distribution shift setting. We contribute a versatile scenario characterization method, a novel scoring scheme for reevaluating a scene using counterfactual scenarios to find hidden risky scenarios, and an evaluation of trajectory prediction models in this setting. We further contribute a remediation strategy, achieving a 10% average reduction in predicted trajectories' collision rates. To facilitate future research, we release our code for this overall SafeShift framework to the public: github.com/cmubig/SafeShift