Well, it seems unlikely it's the menu cache in and of itself because I cleared that database table repeatedly while experimenting with solutions. And it's the same cache whether I'm on the cart page or the checkout page, right? I examined the menu cache serialized data from the {cache_menu} table and found that "cart/checkout/shipping/quote" didn't exist in it whether on the cart page or the checkout page. (This must be because it is entered in the "else" section of the if ($may_cache) clause, and so isn't cached. I think. Don't understand that damned 5.x menu system.)

I must sadly point out, by the way, that my workaround did NOT solve the problem after all. It worked fine for a while, and now the behavior has recurred and won't go away. In other words, my above fix seems to have been ineffective after all, though it's vexing beyond words that it would have worked before and then suddenly not. I can't think of what changed and why it only changed temporarily.