Dice are pretty much synonymous with RPGs, but there are a few rare systems that forego them altogether, like Castle Falkenstein.
What are some other systems that don’t use dice? Are there any that completely remove luck/random chance on “important player actions”?
Sorry for the long reply
Running Amber DRPG as a diplomacy focused story is certainly an option, especially if you’re doing a throne war campaign. There are certainly other setups beyond this (we started immediately following book 5 as grandchildren of Oberon). While every PC and NPC certainly has their own motivations and schemes, there is no requirements as to how political you need to be (Gerard notoriously hated the petty scheming of his siblings; Benedict doesn’t care as long as Amber remains stable).
As for advancement (which I think is what you’re alluding to with ranking change): if you’re following the book’s rules as written, the #1 in each rank for each generation cannot change (Benedict will always lead in warfare). Advancement is also blind, so you will no longer know your stats.
As for why it has run a long time: thankfully, I have a great GM that has crafted a compelling story and wonderful players that agree on the tone of the game and are reliable. It also helps that we started as 100pt characters, so there has been room for growth and development.