Of course, work on a family never centers on one document (or at least should not, if at all possible). In the case of Martha and others, her death certificate should be analyzed in light of other evidence. And it should always be remembered that documents can be incorrect. Her death certificate had information provided from someone outside the family--a prime source of potentially incorrect information.
The 1900 census for Rocky Run Township, Hancock County, Illinois even has the last name as Sargin instead of Sargent.
Martha's 1900 census entry (itself prone to errors) gives a different middle initial for her and a year of birth off from the census, but not radically different.