Frances's findagrave suggests she had five children: Martin Van Buren-1858, Elmer Ellsworth-1859, Wilbur E-1866, William E-1867, Frances Jenny-1870. I don't think that William existed. I think Wilbur is William. There was no Wilbur in the household in the 1870 census, but there was William. Why would Wilbur be elsewhere? I think William only "exists" because of that 1870 census. But until I have evidence one way or another, I'm going to keep them separate entities here. 1860 census lists only Wilbur and not William. I think William, Wilbur, and Willie (from 1880 census) are all one.
Now I've found a Wilbur E buried in Winthrop and one buried in Leeds. Winthrop one was 7 yoa, son of BF Burgess and Frances. Leeds one was married and had a family, also son of BF Burgess and Frances. Perhaps they named two kids the same? Seems a little late (1860s) to have done that, but ya never know. Photo request on findagrave for the Winthrop one has been made.