No. TD status allows a spouse and children to accompany the TN professional and to study, but it does not authorize employment of any kind. He needs his own classification if he intends to work.
TD does not carry a work permission for anyone
Hypothetical example: a Mission millwright assumes he can pick up maintenance shifts once his spouse is admitted as a TN forester. He cannot. TD is a dependant status; it does not become a work permit because the principal's employer is short of tradespeople.
Millwright is not a profession on the USMCA Appendix 2 list, so a TN of his own is not available for that trade. His options depend on his own facts: an employer of his in Canada with a related U.S. entity might support an L-1 transfer after a year of qualifying employment; other classifications have their own rules.
What he should not do is accept cash work or remote paid work performed from the United States while in TD, because unauthorized employment affects future applications for both spouses. Study, including trade courses, is permitted. Hypothetical example: An urban planner expects to begin TN work while her spouse plans to study and later accept a U.S.
research-assistant job. The family review should identify every traveller's intended activity before the principal applies. TD can support accompanying family members and study, but it does not authorize the spouse's planned employment.
The record that settles the immediate point is the spouse's admission classification and I-94, not the principal's offer letter. Sequence the applications so the spouse has an independent work-authorizing status before taking paid duties; unpaid observation should also be assessed carefully when it resembles productive work.