Does a good life require one central purpose?
Yes, in the Objectivist sense—but not in the cramped sense of having one obsessive passion that crowds out the rest of life.
A central purpose is not just a hobby, career ambition, or productivity system. It is the major productive aim that gives long-range structure to a life. In Objectivism, productive work is the central purpose of a rational life: the central value that integrates and helps order the rest of one's hierarchy of values.
That does not mean careerism, workaholism, or treating every other value as subordinate in the sense of being expendable. Health, love, friendship, rest, art, play, sex, family, learning, and beauty can be genuine values. But without some central productive direction, they can become disconnected fragments. A central purpose helps answer questions line "What am I building", "What kind of work gives my life forward motion?", and "How should I organize time, attention, learning, ambition, and tradeoffs?".
This also does not mean everyone needs a grand public mission. Productive work can be great or modest, intellectual or physical, entrepreneurial or domestic, artistic or technical, paid or partly unpaid. The issue is not social prestige. The issue is whether you are using your mind to create, maintain, improve, or build real values in a sustained way.
For fitness and self-development, this matters because a life without direction makes habits feel arbitrary. Physical exercise, nutrition, sleep, and emotional regulation become easier to sustain when they are connected to a life you are actively building. The question is not merely "What routines should I follow?", but "What kind of person and life are these routines serving?".
So the practical answer is yes, a good life needs an integrating productive direction. But it should organize your values, not devour them. If the idea of a central purpose makes you frantic or it seems artificially narrow, the next step is not to reject purpose; it is to define it more honestly. Ask yourself what work, creation, responsibility, or long-range project would make life more integrated, more self-directed, and more fully yours?