Lots of countries (eg. Panama) won’t let you in with less than six months left on your passport. Others are more lenient and won’t take you with less than three months to go.
What happens? Well, airlines might refuse to board you. And there goes your expensive pre-paid vacation. Or your hard-fought-for reservations at that exclusive resort. Or worse yet, your entire two week vacation of baking on the beach in paradise, leaving you stranded instead in the frozen north.
Or suppose you have a non-refundable flight from Panama booked to a country that requires three months on your passport, and you have only two left. The airlines won’t let you board. What are you going to do? Call the US Embassy and demand that they fix it? Sorry, they can’t.
And what happens if you are so distracted by the gorgeous beaches here in Panama that you actually let your passport lapse? Tch. Or you leave your hotel without it and are stopped by the policia for some reason? Tch-tch. Guess what? There is a special “detention center” in Ciudad Panamá for wastrels like you. You can call the Embassy from there and demand that they fix it. They’ll do what they can, but you’ll be “detained” until they get it sorted out.
On the other hand, if you should actually be so unfortunate as to lose your passport or worse, have it stolen, they CAN fix it, and will issue you some temporary ID.
What the Embassy does best regarding passports is to help you renew yours in timely fashion. Give that at least three months to happen. They have to pack up the old one and send it via diplomatic courier to DC, and that particular mail doesn’t go out as often as you might think. So plan ahead. You’ll get temporary ID so you won’t wind up in jail, but it is not useful for leaving the country.
What’s the Boy Scout motto? Be prepared.