In studying the evolution of long term care, it's important to understand the way that hospitals were evolving, since there was some overlap between "old age homes" and "hospitals." Hospitals have some roots in the poorhouse system, just as nursing homes do, since from the beginning of the country many of the poor have been old and sick, and many of the old and sick have been poor. Some of the first hospitals in the country were built on poor farms to house the sickest of the poor population. (Medical History of Michigan, 1930) Where there was no poorhouse hospital, private hospitals received money from the county poor fund to care for the sickest paupers, which often made up a significant percentage of their patients. A few poorhouse hospitals, like Cook County Hospital in Chicago, still exist today as public hospitals.
Nineteenth century hospitals were not places where you expected to be cured, they were places to go when all other options had been exhausted. In the early part of the century medical science was very crude, and often consisted of "cures" like bloodletting, where the doctor cut the veins of the patient to let "bad blood" escape. In some cases the best outcome you could expect was that the "cure" wouldn't kill you. It wasn't until the latter part of the 1800's that researchers began to understand how to deal effectively with illness and disease.
In the earliest years of the country, and in the parts of the country that weren't yet settled, people cared for their sick at home. If a doctor was needed, he would come on horseback or by buggy, sometimes staying overnight. As time went on, some doctors began to board a few of the sickest patients in the doctor's own home. As demand grew, religious and other nonprofit organizations built better facilities and hired additional physicians. Hospitals as we know them today began to emerge in most of the country in the early- to mid-1800's, a bit earlier than the voluntary and non-profit old-age homes. Some early hospitals included care for the elderly as a part of their mission, even building "homes" attached to the hospital where the poor elderly could live. (Medical History of Michigan, 1930)
Unlike the younger, healthier patients, poorhouse patients tended to have chronic conditions that required long term care. People who weren't poor cared for the chronically ill at home. Those who were poor and ill, many of whom were also elderly, often ended up in hospitals for very long periods of time. (Charity Hospital, 1890)