The Family That Dwelt Apart (1973)

Plot Summary

A family of seven lives on a New England island with nobody else. One winter, they got stranded because the bay froze over. This didn't much matter to them, but someone on the mainland remembered them and launched an aerial rescue mission; an unfounded story that one boy had appendicitis results in a larger rescue mission, with the situation rapidly escalating into a major catastrophe all because of the rescue mission.