No. TD status carries no work authorization at all, and that includes remote work performed from inside the United States for Canadian clients. He could study, but any paid work would need a status of his own that permits it.
TD lets a family live and study here, and nothing else
The restriction catches people because the work feels Canadian. If Wendy's partner sits at a desk in the United States and issues invoices to farm clients back home, the work is being performed in the United States and TD does not authorize it. The same applies to occasional consulting, a side business or unpaid work that would ordinarily be paid.
What TD does allow is living with her and enrolling in study. Two further points deserve early attention. The spouse must be a legally married spouse; a common-law partner does not qualify for TD and would need an independent basis to accompany her.
And a child in TD status may attend school but cannot work, and loses eligibility on turning twenty-one. A household that depends on two incomes should test the other spouse's own options before relying on TD.