Effective stroke triage in emergency settings often relies on clinicians' ability to identify subtle abnormalities in facial muscle coordination. While recent AI models have shown promise in detecting such patterns from patient facial videos, their reliance on real patient data raises significant ethical and privacy challenges -- especially when training robust and generalizable models across institutions. To address these concerns, we propose SafeTriage, a novel method designed to de-identify patient facial videos while preserving essential motion cues crucial for stroke diagnosis. SafeTriage leverages a pretrained video motion transfer (VMT) model to map the motion characteristics of real patient faces onto synthetic identities. This approach retains diagnostically relevant facial dynamics without revealing the patients' identities. To mitigate the distribution shift between normal population pre-training videos and patient population test videos, we introduce a conditional generative model for visual prompt tuning, which adapts the input space of the VMT model to ensure accurate motion transfer without needing to fine-tune the VMT model backbone. Comprehensive evaluation, including quantitative metrics and clinical expert assessments, demonstrates that SafeTriage-produced synthetic videos effectively preserve stroke-relevant facial patterns, enabling reliable AI-based triage. Our evaluations also show that SafeTriage provides robust privacy protection while maintaining diagnostic accuracy, offering a secure and ethically sound foundation for data sharing and AI-driven clinical analysis in neurological disorders.