Very little chance of a hoax in my mind. If the coast guard heard the signal (as they claim) rather than "received a report" then they know where the signal came from. They have good quality radio direction finding equipment. If the call came from SF bay, or from a marina, then it would be a different story.
As to the other reasons it may be a hoax, I would say that the same person who doesn't have a raft, and doesn't have a GPS IS the same person who might also might go out in the winter (not knowing better) and might be in a boat that may not be sound.
God be with them is right. This sport can be very safe, or very dangerous, and a few key elements separate the two.
I do apologize for bringing up the possibility that this may be a hoax. As bad as that would be, it would actually be preferable in this situation, considering the conditions out there and what little safety equipment these folks seem to have with them. Considering that no one has reported this family missing does make one wonder, especially since a member of another family (the cousin/nephew) was also on board. If they were making a longer trip, it would be more understandable since they may not be expected to arrive anywhere yet.
As for remaining calm during the radio call, I made the comment because I personally would not sound that calm when faced with getting into those seas with a cooler and a life ring and two small children. The offshore weather report for that day was seas 14-20 feet, subsiding to 9-15 feet, with a gale warning. I do admit that others may be calmer, including my husband -- who, like most good captains, keeps his head during stressful situations.
The fact that they have a bearing on the radio signal does make it seem more probable to this is unfortunately a bonafide tragedy and not a hoax. I do have a question about the bearing of the signal. Can the USCG determine the distance or just the bearing? I'm guessing only the bearing? Thanks for any clarification.