After a little more digging, I discovered this:
Items cannot be added from the category pages, the shopping cart buttons do not work.
Hwever, items can be added to the shopping cart with the "add item to cart" button on each item's individual page.
Checkout appears to work normally from that point.
So the question is, what would disable the quick add button on category pages when the site otherwise is working?
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OK, here's more: after going to an item page and then back to a category page, the cart icons reappear on the category pages.
However, if I leave the site and close the tab and then reload the home page, the icons do not appear.
I am stumped.