The bureaucracy has tainted both the words and the thing, since any sort of dynastic legacy made now is likely due to a particular family's capacity to work the existing system. And the words- what a mess they've made of rights as any sort of meaningful thing, and how successful they have been in demonizing anyone that might want a dynasty.
Additionally, I doubt the ancients would necessarily agree with a right to dynasty. Indeed, those who already had a dynasty, would no doubt use whatever power or political means necessary to keep themselves at the top. I wonder if it was as bad as now though, since those seeking to improve their families' lot in life could, if they were successful, marry into the local aristocracy.
A dynasty- a functioning intergenerational estate, which would include businesses and necessitate good administrative qualities- is really the only sign the rest of us in this world have as a clue that someone might make a good leader. It is not something that provides 100% proof, and obviously many myths have been made about families wherein the 3rd or 4th generation ends up destroying whatever their ancestors made- though a lot of the examples of this comes from the post-bureaucratic age, when people are miseducated and the bureaucracies themselves are likely to notice your estate and think of it as a nice large pile of assets to steal.
Destroying humanity's dynastic impulse means they've largely successfully destroyed examples of good governance, or at least hidden them. Additionally, we are continually encouraged to use whatever wealth we might build up for ourselves, to focus on short term consumptive goals. Our house, our family, our lineage- it once made up a large part of this thing we called identity. What is identity now? Often a self-destructive delusion.