The US founding fathers implemented the Quranic injunction that does not sanctify lineage or the system of monarchy and family rule.
This is exactly what goes against the Shia concepts of rule. Ponder over the Quranic verse that clearly says:
33:40 Muhammad is not the father of any man among you, but he is God’s Messenger and the closing seal of all Prophets. God is Knower of all things.
It means that God had forewarned about family rule problem cropping up later. The Messenger's sons would not have died if God had wanted the lineage and authority to continue. This is something that Shias refuse to understand.
Islam was a movement and movements have leaders and not there is no inheritance of leadership. The close associates and others were aware of the fact that the Messenger had no surviving son. The verse clearly warns against family rule.
The Americans had learnt a lesson from Europe and did not adopt monarchy or family rule