I just had a very quick search for Joseph, John and J F Rutherford on the Ancestry website.
I only pay for access to UK records, so unfortunately not able to check US censuses, but to give an idea, listed in the 1871 UK census, there are 17 different Joseph Rutherfords born around 1869 and 64 John Rutherfords . If anyone has access to the US Ancestry website, you will very likely find similar results, including the Joseph and John in question.
The free Latter Day Saints website lists 71 Joseph Rutherfords alive in the 1880 census, including 5 in Missouri.
It's to be expected that several related people who shared the same surname and initials would have lived in even the tiniest of towns or villages. Even a few who also shared the same forename and surname.
Generally people tended to stay in the area where their family was based, so it is not uncommon to find two cousins born the same year with the same name, because two married brothers chose to name their respective baby boys after their own father, (the babies' grandfather). Include the extended family with uncles, second cousins etc, especially with popular names like Joseph and John and you have a high likelihood of finding a namesake even in a one horse town.
(Incidentally, I came across the distribution map of the surname Rutherford and it apparently originated in the borders of Scotland.)