Backpack is ofcourse the most important part in the checklist and that is what we shop first. Walmart usually has a great collection of backpacks.
Because hygiene became an important part of everyone's life so, carrying a hand sanitizer is preferred everywhere. So I think you should include that in your checklist as well.
Shop for school outfits but don't buy for entire year in one go. Wait for season sale, that way you can save some money. Also Shein has amazing collection.
you need to take similar approach for school supply as well. Just buy 1 month school supply and then wait for clearance or sale. you will get same stuff in less price.
You also need shop for your kid's favorite food items, lunch boxes, water bottles etc. There are couple of things that you can buy in bulk to avoid daily shopping.
I am sure you might have covered all the previous items in your list. Vaccination has become an important requirement after pandemic. So you need to make sure that your kids are covered.