TD status does not authorize employment, and working from inside the United States for a Canadian employer is still work performed in the United States. The spouse's situation needs its own analysis before he plans on a U.S. desk.
Remote work is still work where the person sits
TD is available to the spouse and unmarried children under 21 of a TN professional, and it allows study but not employment. The common misunderstanding is that a paycheque from a Canadian employer, deposited in a Canadian bank, stays outside U.S. rules.
It does not; what matters is where the work is physically done. Hypothetical example: a Boischatel school-board maintenance coordinator wants to keep his role by handling scheduling and vendor calls from the couple's New Hampshire apartment. That plan should be reviewed with counsel before he tells his employer anything, because the answer may be that he needs his own route, or that his role cannot be done from there.
Children can enrol in school on TD. Everyone's status ends with the principal's, so the household should also record what happens if the inspection contract is cut short.