I would love to have an official 5e version of Birthright.
To be honest the domain rules for AD&D were almost completely distinct from the D&D rules so I’m pretty sure you could port them over wholesale.
Create bloodline powers that are equivalent to feat in strength and allow players to take them as such throughout their career.
Create a feat called Bloodline equivalent to a level one feat that lets you draw on your regency and gives you a minor power. Lucky would be a good base.
For the purpose of Domain Management keep the score generation the same as in the 2e version. Or have a characters starting Bloodline Strength is equal to 10 + 2d6 + proficiency bonus + the total bonuses of the characters stats. Bloodline strength can be increased by in game activity anyway.
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BLOODLINE
1st-Level Feat
Prerequisite: None
Repeatable: Yes
You have inherited a bloodline from one of the old gods, granting you the power to rule the land. This power can also be harnessed to strengthen your blows or protect you from harm. Pick one of the gods whose bloodline you hold. This cannot be changed, and gain the benefits below.
Bloodline Points. You have a number of Bloodline Points equal to your Proficiency Bonus. You can spend the points on the benefits below,and you regain your expended Bloodline Points when you finish a Long Rest.*
Advantage. Immediately after you roll a d20 for a d20 Test,* you can spend 1 Bloodline Point to give yourself Advantage on the roll.
Disadvantage. When a creature rolls a d20 for an attack roll against you, you can spend 1 Bloodline Point to impose Disadvantage on that roll.
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That’s pretty much the rules ported across.