Use the store’s boxes.
That’s the way I do it at Adli’s. I keep 4 or 5 boxes from the store in the trunk, load them from the cart and use those boxes to carry the groceries into the house. No bagging necessary and quicker too.
Here in the Philippines the supermarkets ask you if you want plastic or box. When I do my monthly shopping we ask for the box for most of out purchase but plastic for the items that can get squished, (bread, chips etc.). Meats are placed in plastic bags and, (since I live an hour away from the city and decent supermarkets), I bring an ice chest and buy a bag of ice to keep the meats cold until we get home. The box is closed and secured using a thin plastic string, called straw, and is very good at keeping the box from opening from the bottom.