Would they do that for an international departure? They know where you’re flying, and I’d think they’d just tell you to stop being an idiot and show them the passport you obviously must have. But policies can be weird, so maybe not.

Yes, that's what I said above. The US government doesn't give a toss, but the airline has to fly you back if you're refused entry at your destination, so they will do their best to ensure you have the documents you need.