New York magazine not merely repeated the story, but bothered to interview Harrison Tyler, one of the grandsons. Not only is he still alive at 83, he lives at the Sherwood Forest Plantation in Virginia -- home of the president from 1842 to his death in 1862. The other grandson, Lyon Gardiner Tyler, lives in Tennessee.
For the record, Harrison and Lyon are the sons of Lyon Gardiner Tyler (August 24, 1853--February 12, 1935), who was the fifth child of John Tyler and his second wife, Julia Gardiner Tyler. The elder Lyon was born when his father was 63; Harrison and Lyon were born when their father was in his 70s. This full interview is here.