I take it from your reference to BMO that you are Canadian? There aren't as many options for crazy credit card deals for folks outside of the US.
Typically, however, what folks do is earn airline miles with a specific airline (say, Air Canada) and then use those miles to purchase "award" tickets from the airline in question. A different option, similar to what you seem to be talking about, is getting a card that offers a rebate on purchased travel using that cards "miles" or "points" - these would be Chase bank's Ultimate Rewards, American Express' Membership Rewards, or Barclay bank's "miles" from their Arrival cards. In nearly every case, getting the rebate is not a good deal, while using the miles to buy award seats with an airline is a good deal. Some of these "miles" can be transferred to airlines to use there (e.g., Ultimate Rewards can be transferred to United Airlines to use for United award tickets).
Your second issue is that you're trying to buy tickets now for this winter, just like everyone else. To get the best travel deals you often have to purchase 10 months-1 year in advance. For example, I used miles in January to get tickets for September travel.
To give you a sense of what "normal" is, it costs 30,000-35,000 airline miles for 1 person to fly round trip in coach from Canada to the Caribbean on most US airlines. When you say "the family" if you mean 3 people, you'd need 90,000-105,000 airline miles for those 3 tickets.
Edited to add:
https://canadiankilometers.boardingarea.com/and
https://pointshogger.boardingarea.com/ have a Canadian take on earning airline miles.
