Yes. An L-2 spouse is employment-authorized incident to status, and that authorization covers remote work for a foreign employer performed in the United States as well as U.S. employment. Tax and payroll arrangements are a separate matter.
L-2 authorization covers the remote arrangement TD would not
This question is the mirror image of the TN household's problem. A TD spouse cannot work at all while physically in the United States, including remotely for a Canadian employer. An L-2 spouse can, because current policy treats L-2 spouses as authorized to work incident to status, with the annotated admission record serving as evidence.
So the engineer's husband may continue his Canadian employment from their U.S. home, take U.S. work, or both.
What changes is not immigration but administration: his Canadian employer will need to consider payroll, withholding and the tax residence of an employee living abroad, and he should take advice on both countries' tax treatment. Children in L-2 status may attend school and may not work, and their status ends at 21 or on marriage. The family's status depends on her L-1B continuing and is limited to the same period.
If the household is weighing routes, the spouse's ability to keep his existing job is often the fact that makes L-1B preferable to a faster TN application through a different employer.