Artificial intelligence (AI) mental health chatbots have become popular alternative tools in mental care by enhancing the accessibility and scalability of psychotherapy. However, integrating human-centric AI technology and mental health services poses challenges, as safety, reliability, and trust issues have not been fully addressed. This study conducted a case study with two AI mental health apps, Wysa and Youper, to explore the implications of using AI mental health chatbots as therapeutic tools to provide accessible, immediate, and personalized mental health support. The study emphasizes the importance of a human-centric AI approach by examining whether these apps are ethically aligned and reliable enough to address diverse symptoms effectively. Through a comprehensive analysis, this paper outlines design suggestions for developing human-centric AI mental health chatbots while also examining potential challenges and limitations, such as maintaining partnerships with human therapists, ensuring privacy, and the necessity for continuous improvement through feedback loops. With careful design and ethical consideration, AI mental health chatbots can significantly contribute to the mental health field by offering solutions to reduce the burdens of manual tasks of human therapists and providing accessibility to patients.