Recent advancements in HCI and AI research attempt to support user experience (UX) practitioners with AI-enabled tools. Despite the potential of emerging models and new interaction mechanisms, mainstream adoption of such tools remains limited. We took the lens of Human-Centered AI and presented a systematic literature review of 359 papers, aiming to synthesize the current landscape, identify trends, and uncover UX practitioners' unmet needs in AI support. Guided by the Double Diamond design framework, our analysis uncovered that UX practitioners' unique focuses on empathy building and experiences across UI screens are often overlooked. Simplistic AI automation can obstruct the valuable empathy-building process. Furthermore, focusing solely on individual UI screens without considering interactions and user flows reduces the system's practical value for UX designers. Based on these findings, we call for a deeper understanding of UX mindsets and more designer-centric datasets and evaluation metrics, for HCI and AI communities to collaboratively work toward effective AI support for UX.