No. A child over twenty-one has no derivative status at all, and even under twenty-one a dependent child may study but not work. He would need a status of his own. An E-2 spouse is employment-authorized incident to valid E-2 spouse status, but that authorization does not extend to a child.
An adult child is a separate application
Derivative status covers a spouse and unmarried children under twenty-one, and it ends at that birthday regardless of whether the family is settled. An adult son who is to manage the shop needs his own basis to work in the United States. One possibility is a treaty investor employee classification, which requires that he share the investor's nationality and be employed in an executive or supervisory role, or possess skills essential to the enterprise; a genuine general manager role can fit, and it is a real petition with its own evidence rather than an extension of his parents' status. Another is a different employment classification if his qualifications support one. The point to absorb early is that his role must be planned and documented as a separate case, on its own timetable and at its own cost.
Hypothetical example: an indoor climbing gym will employ a couple and their nineteen-year-old daughter as its marketing lead. A first review produces a family chart with each person's age, intended job and independent eligibility. The daughter's birth record settles the derivative relationship, but it does not give her a permanent solution after she turns twenty-one. If her role is essential to the business, the parents must decide whether a separate employment classification is realistic or whether the staffing plan should use another worker. Leaving that decision to the renewal stage can disrupt both the business and the family.