It was Sinclair Fox, Junior, whom the 1860 census shows in the the "County Infirmary for the Poor." He was there because of "intemperance," the politically correct Nineteen Century word for alcoholism [he died there in Nov 1861].
Not all residents were old folks; some were identified as insane or afflicted with various diseases. There also was a sprinkling of young widows with children.
If Sinclair Fox, Sr., was around in 1860, he definitely would have been the oldest person in Ashtabula County as he claimed Aug 1758 as his DOB [on Rev War Pension App].
Sinclair Jr. is a bit of a mystery. As far as I know, he had one short-lived marriage and no children. He seems to disappear from all records shortly before 1840 and doesn't reappear until the 1860 census. By then, most of his siblings had either died or left Ohio. One can only speculate why those who remained didn't want to take care of him.