The use of Artificial Intelligence (AI) in healthcare, including in caring for cancer survivors, has gained significant interest. However, gaps remain in our understanding of how such AI systems can provide care, especially for ethnic and racial minority groups who continue to face care disparities. Through interviews with six cancer survivors, we identify critical gaps in current healthcare systems such as a lack of personalized care and insufficient cultural and linguistic accommodation. AI, when applied to care, was seen as a way to address these issues by enabling real-time, culturally aligned, and linguistically appropriate interactions. We also uncovered concerns about the implications of AI-driven personalization, such as data privacy, loss of human touch in caregiving, and the risk of echo chambers that limit exposure to diverse information. We conclude by discussing the trade-offs between AI-enhanced personalization and the need for structural changes in healthcare that go beyond technological solutions, leading us to argue that we should begin by asking, ``Why personalization?''