My DW and I vacation in a housekeeping cabin in the eastern Sierras that is, for how we like to cook, poorly equipped - basically the cheapest things they could buy at a Walmart. New management has helped some but it still isn't to our liking It sounds very much like your AirBnB kitchens. Because we travel in a pickup truck with a camper shell space is not an issue for us. We carry all of the pots and pans we will need for our menus, plus the 2 large and 1 small tubs full of items of our preference.
I would suggest a roll of heavy-duty foil. Useful for many things.
Here is a picture of some of the utensils we carry in case anything pictured suggests something to you. I carry a separate knife roll but you already know about your knife needs.
I envy your ability to take your children on trips like you describe. Now we have grandchildren in that age range.