As an unmarried child under 21 he may accompany the family and study, but he may not work, paid or unpaid, at the inn. At 21 his derivative status ends. If he wants to work in the business he needs a status of his own, and for a family enterprise that is often difficult.
A child's derivative status is a countdown
E-2 dependants are the spouse and unmarried children under 21. The spouse is employment-authorised incident to status and can work at the inn or elsewhere; the children are not, and helping at the front desk without pay is still employment for immigration purposes. Hypothetical example: the Lake Country couple's son plans to spend two years at the inn learning the trade before university; he can live there and enrol in a local college on his E-2 derivative status, but he cannot take shifts, and on his 21st birthday his derivative status ends regardless of what the family has planned.
Options at that point include a student visa if he is enrolled, or an E-2 in his own right if he later invests substantially himself, which requires his own funds and role. If the son is expected to take over the inn one day, the family should build the succession around a status he can hold independently rather than around his derivative status.