Recent large language models (LLMs) have advanced the quality of open-ended conversations with chatbots. Although LLM-driven chatbots have the potential to support public health interventions by monitoring populations at scale through empathetic interactions, their use in real-world settings is underexplored. We thus examine the case of CareCall, an open-domain chatbot that aims to support socially isolated individuals via check-up phone calls and monitoring by teleoperators. Through focus group observations and interviews with 34 people from three stakeholder groups, including the users, the teleoperators, and the developers, we found CareCall offered a holistic understanding of each individual while offloading the public health workload and helped mitigate loneliness and emotional burdens. However, our findings highlight that traits of LLM-driven chatbots led to challenges in supporting public and personal health needs. We discuss considerations of designing and deploying LLM-driven chatbots for public health intervention, including tensions among stakeholders around system expectations.