I emailed Matthew Simon, the current owner of the house that the 19th century John Bethel built, and he wrote back and quoted from a post on the "Straight Dope Message Board, as follows (I've corrected some typos that were in the original post):

John Bethel was a sea Captain from Govenours Harbor, Eleuthera, The Bahamas. The "Sloop John B" was an actual boat (sloop) that Captain John Bethel built in the front yard of his house. He then rolled it down hill on logs and launched it into the harbour. Captain Bethel used the Sloop John B for many years in commerce, both in The Bahamas and along the eastern seaboard of the United States, from Key West all the way up to Newfoundland. At some point the boat hit a reef and sunk. Someone wrote a poem, another put that poem to music. That song became a sailors sea song and was adapted widely and sang by many sailors. for many many years, depicting the drama on a typical sea voyage ... In June of 2014, I stayed in the Sea Captains House, which is over 300 years old. This story is common knowledge on Eleuthera.

Go to VRBO and search for The Sea Captains House, Eleuthera. Angelicka Bacchus, a realtor and restaurant owner on Eleuthera can tell you more details.

Matthew then added:

Except that the sloop sank in Governors Harbour (where there are no reefs) and the wreck was a landmark in the harbor for many years. John Bethel lived in the late 19th century. He was perhaps descended from the 17th century John Bethel, but not the same guy : )

So it seems to be pretty certain to me that the Sloop John B (both the boat and the song named after it) was named after its captain, John Bethel, who lived in the mid-to-late 19th century; and that the song dates from around that time.

As the boat was sunk in Governors Harbour, which is on a different island from Nassau, the note in the DT version of the song is probably wrong when it says: "This is a theme song of the Nassau waterfront, telling the story of a little sloop and the wild party which went on the night she was sunk."

I'm beginning to think that the reference to "the wreck" of the Sloop John B in the song's title must just be a reference to the fact that everyone in the Bahamas at the time the song was written knew about the wreck of that boat, as it was a landmark back then - but the song is not actually about the boat being wrecked, (or even about what happened the night before it sank) - but rather, it's just about the behaviour of it's crew, who according to Blind Blake "were in the habit of getting notoriously merry, whenever they made port".

Dave