As I understand it, the popes in this period generally came from powerful families. Who within the family would choose to pursue becoming a cardinal or pope? If it was not the head of the family but someone else how would the power balance within the family look like if he became a cardinal or pope? And if it was the head of the family who would carry on the family name since he were not allowed to produce heirs?
From what I know of renaissance era popes, family lines could be carried on both directly and indirectly. Many popes were cousins, nephews, and other relatives of previous popes, usually of old and (varying) noble families. Perhaps the most notorious pope, Alexander (Rodrigo Borgia) is a great example of this, having had an uncle who was pope, as well as an example of popes carrying on family lines regardless of vows. The Borgia pope had three illegitimate children who inherited and married into other noble and royal lines, carrying on the family name and legacy. Though Rodrigo Borgia was on the extreme end of the example (because he acknowledged his children) he was hardly the first pope in Italy to father children. Here on Wiki is a comprehensive list of popes rumored/proved to be sexually active/have issue, which includes dates of potificate and whether they had issue before or after papal reign. It's not hard to imagine from the workings of the time that anyone who had a pope in the family would generally defer to him as head of the family, with all the power and prestige that came with being pope. Popes were immensely powerful and the favors and advantages that came with being in his good graces would make him virtually if not actually head of any family.