No, there are also mid-size and large-scale family farms. Many of those are owned by corporations, but that corporation is owned by a single family so it counts as a family farm. "Small" is defined as gross income of $250,000 or less.
A gross income of 250,000 isn't a small farm, it's a hobby farm. The average margin for farms is 15%, and smaller farms are worse than large farms. A $250,000 gross likely means a <$20K net, and you can't support a family on that in America even if you grow your own food.
Worse than the small margin is the variability. A good year can have a 50% margin and a bad year a -25% one.