It was the sort of miracle cure that the board of a rural Oklahoma hospital on the verge of closure had dreamed about: A newly formed management company promised access to wealthy investors eager to infuse millions of dollars.

The company, Alliance Health Southwest Oklahoma, secured an up to $1 million annual contract in July 2017 to manage the Mangum Regional Medical Center after agreeing to provide all necessary financial resources until the 18-bed hospital brought in enough money from patient services to pay its own bills.

But about a month later, hospital board members were summoned to an...