With primogeniture, there is lots of motivation to kill your older brothers and take over the throne. This causes families to self-destruct once they reach power.
If everyone in a country was absolutely obsessed with the idea that the human pinkie finger was a symbol of a leader's connection to god and was required for him to be a good ruler, this whole problem would be solved. The oldest brother could simply have his younger siblings' pinkie fingers cut off, and let them live.
This would mean that ruling families would be much more free to have big families and expand their power even more.
The highest office would require a fully complete pinkie finger on both hands. Being a high councillor would require all but the last joint. Lower roles would require less. There would be a coming of age ceremony where how much of their finger would be allowed to be kept. Occasionally bandits would come in from other cultures with full fingers and everyone would be amazed.
This would work well in isoluation - murders of family members to prevent them from competing with you would be much less frequent; but one bad effect would be that in some cases it would be hard for people to advance. Plus, in isolation, soldiers having missing fingers would put this culture at a disadvantage in wars; occasionally fully-fingered barbarian tribes would invade and wow everyone.