Whilst I agree, to be fair, a passport is usually only needed when entering a country, not leaving one, right? Under the cited rule, the US needs to allow you to leave, not help you in entering some other country.
I have yet to leave a country (well, a state technically) without having to show a passport - with the exception of the Schengen area.
It is quite difficult to leave a country without simultaneously entering another
You generally do present your passport when leaving. Most places you get an exit stamp (which matches your entry stamp). They usually confirm things such as not overstaying a visa.
ex:
overstaying in Thailand results in a on-the-spot fine
China lately has exit checks when traveling to SEA (they try to intercept people traveling to scam centers)