There were no beeping machines or blinking lights or scurrying medical residents. A volunteer circulated among the patients like a flight attendant, making soothing conversation and offering reading glasses, Sudoku puzzles and hearing aids. Above them, an artificial sun shined through a skylight imprinted with a photographic rendering of a robin’s-egg-blue sky, puffy clouds and leafy trees. Ms. Spielberger, who is in her 80s, was even getting into the spirit of the place, despite her unnerving condition. “It’s beautiful,” she said. “Everything here is wonderful.” Yet this was an emergency room, one specifically designed for the elderly, part of a...