As health AI tools enter clinical spaces, how they are perceived by providers in safety-net environments shapes whether their benefits reach underserved populations. We examined the degree and drivers of healthcare providers' trust in health AI, whether that trust moderates perceived benefits for safety-net populations, and how providers balance benefit against concern across four specific hypothetical AI tools. We used a two-study mixed-methods design: 18 semi-structured interviews with healthcare practitioners and research leaders, followed by a survey of 229 Texas healthcare providers incorporating four vignettes describing hypothetical AI tools, each rated for appeal and concern and followed by open-ended explanation. Responses were thematically coded and stratified by self-reported trust in health AI (47.6% high trust, 52.4% low trust). Low-trust respondents most often cited building trust (50.5%) and data bias and accuracy (33.0%) as drivers of their current trust level, while high-trust respondents more often cited efficient and effective care (23.5% vs. 3.7% low-trust, p