The political dynasty is a bit of a contradictory phenomenon. In many republics some families manage to put two or more of their members as a national leader, as if it happens in a kingdom ruled by hereditary succession. A republic itself is essentially formed by a goal to put an end to feudalism culminated in the form of a monarchy, which monopolizes national power for their own family.
Today, the idea that the king is the divine representative of God has long been replaced by the belief that all humans basically have the same right to rule. In general, many nations do not want to be ruled by a person or persons who obtained his position solely because he is a child, niece, daughter or even grandchild of the previous ruler.
At the same time, many of us enjoy the story about kingdoms, kings and princesses in our childhood. That embedded memory makes the most of us like to observe powerful families just like we observe celebrities. People in the United States who are historically anti-monarchy turned out to be fond of Kennedy Clan.
The natural tendency of many people to love stories about monarchy, even legends or myths of monarchy, is also a key factor to make persons from such prominent families to win the competition to gain power. At very least, they are more recognized by the public, compared to those who come from ordinary families.