According to the page which you referred to
it states
Originally Posted by ca
Along with your work permit, you need:
•a valid passport and
•an eTA (if you plan to travel to Canada by air)
and then goes on to say
Originally Posted by CA
If you received your initial work permit before August 1, 2015, you need to apply for an eTA if you haven’t already done so.
If you received your initial work permit on or after August 1, 2015, you already have an eTA. We automatically issued it to you when we approved your initial work permit. It’s valid for 5 years or until your passport expires. See your letter of introduction for details.
Which does say to refer to letter of introdcution for detauls
Are the ETA details on the letter of introduction?
If so, then it would seem that BA was correct that it needs the ETA