Generative artificial intelligence (GenAI) and large language models are rapidly entering mental health research and service delivery, yet their dominant use remains symptom-centric, emphasizing screening, classification, triage, and risk detection. For older adults, mental health is often inseparable from existential concerns: loss of social role, disrupted continuity of self, loneliness, diminished dignity, and questions of legacy. This perspective argues that GenAI should not be conceptualized as an autonomous substitute for clinicians, nurses, social workers, or family caregivers. Instead, it may be better understood as an interactional infrastructure for meaning-centered care in later life. Drawing on meaning-centered psychotherapy, dignity therapy, life review, gerotranscendence theory, care ethics, and implementation science, we propose a Sensing-Narrating-Connecting-Governing framework. In this model, multimodal AI systems help detect existential and relational cues, support life-review conversations, co-construct dignity-preserving narratives, connect older adults with human care networks, and operate under explicit safeguards for privacy, hallucination, dependency, crisis escalation, and cultural adaptation. The proposed framework shifts evaluation from model performance alone toward existential well-being, dignity, continuity of self, therapeutic alliance, equity, and workflow integration. We conclude that GenAI may contribute to public mental health only when deployed as a bounded, human-supervised, culturally responsive layer of relational augmentation rather than as a replacement for human presence.