Only a few private hospitals have survived in Brooklyn neighborhoods like Bedford-Stuyvesant, Brownsville and Bushwick to serve poor patients. Now all are in such dire financial shape that a small group of veteran healthcare planners appointed by Gov. Andrew M. Cuomo is debating last-ditch measures to save them. For decades, the fallback solution in American cities has been to close such hospitals. But one of the actions being considered by the group may be even more radical: expunge the hospitals' debt of more than $1 billion, partly at taxpayer expense, and then let large for-profit companies take over the facilities and restructure patients' care. Experts say what ultimately becomes of the hospitals could make them a model, or a disastrous experiment, in the delivery of healthcare to the poor.