Sorry, I should have updated this thread.
I went to my home country shortly after this whole fiasco, to ring in the new year with my family. A few days after, we decided to all travel to Canada together this time, and on a different airline from before. Guess what? We all arrived in Canada with no issues.
I made sure to go to the airport fully equipped with every document possible: the family's written travel authorizations (again), proof of my residency in Canada, work permit, employment status, job letter, Canadian driver's license, insurance cards covering my family, printed copies of the IRCC pages with all the travel rules and exemptions, etc. You name it. It turns out that I didn't really need to show much; the airline supervisor saw that I was well prepared with everything needed to get my family into Canada successfully. The first thing I even told the supervisor was that my family's travel is essential at this time and that they're exempt from the travel restrictions.
The main takeaway I got from this is that once all the relevant authorizations/approvals are obtained, it's likely a lot more effective if the family member actually travels to Canada with the Canadian resident rather than alone. I would suspect it's much harder for airline staff/CBSA agents to be separating families. However, I can't say for sure that luck didn't play a part in our successful trip.