That's mostly because transport companies have to pay to ship you back if you get turned away at the border, so they will want to see your permission to enter your destination country before you leave. I've traveled internationally a fair bit and I've never had to show my passport to government officials when leaving the US.
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.
