Not long after Beau Braden moved to southwest Florida to open a medical clinic, injured strangers started showing up at his house. A boy who had split open his head at the pool. People with gashes and broken bones. There was nowhere else to go after hours, they told him, so Dr. Braden stitched them up on his dining room table. They were 40 miles inland from the coral-white condos and beach villas of Naples, but Dr. Braden said that this rural stretch of Collier County, with tomato farms and fast-growing exurbs, had fewer hospital beds per person than Afghanistan.So when he proposed starting a 25-bed rural hospital to serve the 50,000 people who live in the farming town of Immokalee and the nearby planned community of Ave Maria, people rallied to the idea. They envisioned a place where mothers could give birth and sick children could get 24-hour help — their own novel solution to an exodus of hospital care from rural America.But then this summer, a larger hospital in Naples derailed those plans by asking the state to deny the proposal, saying that the small, rural hospital would siphon away patients and revenue. The move has upended people’s hopes around Immokalee and delayed any plans to start building the hospital for months. Maybe for good.