Utter nonsense. My friend just got back to the DR from NYC, using an expired drivers license and a copy of his birth certificate. No hassle going in either direction. Well one thing: the migracion on exit tried to charge 2000 pesos for overstaying his tourist card about 2 years but that is not the right fee and he refused to pay it and they backed down to the amount on the signs.
The minute one tourist with a library card and a bus pass tries to leave and gets sent to the Consulate (not consultant) they say "there have been cases". Millions of travellers ,especially charters would be compromised and the DR ain't stupid. What you do encounter are people with similar features trying to get out using relatives' ID, hence this sort of double confirmation happening. And by bet rarely and my bet is when and if it does there is a game afoot.
All that aside, I agree that it makes much more sense to have and use a valid passport when travelling outside your home country. One good reason is any sort of plans to exit in a hurry via other than the USA would fall apart without one. No trips via Panama, Cuba, etc., in an emergency. Canada is cool with US residents and non-passport travel though.