Armstrong alerted the U.S. Coast Guard of their problems. However, the Coast Guard told the couple they were too far out to be rescued.
That's hooey! This is what I believe happened. The sailors contacted the Coast Guard to report their situation. When their position was plotted, the Coast Guard realized that they were well out to sea. At that point, the only option for the sailors is to be removed from their boat by another vessel. When they realized that they'd have to leave their boat behind, they decided to wait it out or arrange for a towing vessel. The bill for a tow boat to come that far out probably exceeds the value of the sailboat.
Finally, when the reality of their situation sunk in, they called the Coast Guard and said they'd like to get off. At that point, an AMVER vessel was most likely diverted to take the crew to safety.
The US Coast Guard is responsible for Search and Rescue in a chunk of the Atlantic that extends out to 040 Degrees Longitude. The area is over 4 million square nautical miles. No matter where you are in that chunk of water, the CG will make every effort to assist you.