What you described is what my family did. We wanted to test the Y-chromosome of my great-grandfather and the mtDNA of my great-grandmother (my mother's paternal grandparents). My great-grandparents had 6 sons and 5 daughters. 5 of the 6 sons had children, and 2 of their 5 daughters had children. As a result, there are many males in the family who are a direct paternal link to my great-grandfather (father to father to father...to my great-grandfather); therefore, they carried my great-grandfather's Y-chromosome. However, a grandson was selected to be tested, who was one my uncles (Mom's oldest-living brother). The test results provided us with the paternal African ancestry of my great-grandfather (through his direct paternal line)...giving my entire family an African identity through him.
As in the case of the mtDNA of my great-grandmother, as mentioned earlier, two of my great-grandparents' five daughters had children. Those two daughters passed on the mtDNA (from my great-grandmother) to all of their children. We tested the mtDNA of one of their children, who was a son, 90 years of age.