A Presidential Coincidence
Students of human nature should be able to work this out!
Suppose that the last of the five presidents mentioned had not died on July 4th. People always want 'interesting facts' to sound as remarkable as possible, so the statement would be that three out of the first four Presidents had died on the same date. We can also be confident that the sixth president - another Adams - didn't die on July 4th or this would have been incorporated into the popular anecdote.