No. A qualifying L spouse may work incident to valid status with appropriate evidence. The title or credential of the proposed job is not the status record.
Keep employment description and permission separate
Eligible dependents include a spouse and unmarried children under 21; children receive no spousal work benefit. Assess the actual person’s status rather than treating the principal’s transfer paperwork as permission for everyone.
Practical steps still matter — employers will need to see documentation acceptable for employment verification — so the family should raise this early rather than assuming a new employer will understand the position. The spouse should still confirm which document a U.S. employer will accept as proof, since employers verify work eligibility from specific records and an admission stamp annotated for a spouse is what USCIS currently treats as evidence. A spouse's employer, a school, and a landlord will each want an answer sooner than the case can give one, and a household that has agreed its own fallback positions in advance answers those questions calmly rather than under pressure. Confirm each dependant’s own status and intended activity separately, including anyone who plans to keep working for a Canadian employer remotely, since that arrangement raises questions worth resolving before departure rather than after.