... I propose a hybrid. Keep the Civ5 unique traits that is totally unique and crafted for each Civ.
But also add a special 'leader' trait that pulls from a pool of traits. This leader trait can be 1 trait or a combination of 2.
Each civ should also have a more formalized 3 unique things, it can be a combination of UU, UB and UI
So, basically, you would add a 'trait' that is one more limiting factor that encourages you to play a civilization spanning thousands of years all according to the attributes of a single leader.
Bald-R-Dash
Call them Uniques or Traits or Cornucopia of Critical Colossal Combinations, but let them be selectable based on the civilization's progress through that particular game:
Mongols on a set of islands do not have to have Keshiks that they have no room to use.
You can play Germany as the Intellectual-Scientific German of the late 19th century, or the expansionist Germans of the Roman-Opposing tribes or the early 20th century - depending on the game and your preference.
Each civilization should have a number of Uniques that are peculiar to that civilization, based on its 'real' history - but the gamer should NOT be forced to try to try to use them in a game in which the 'real' history does not in any way apply or occur. For that, there should be a number of 'generic' abilities that can be chosen based on the actions in the game, combined with civilization-particular uniques or traits that
may be adopted if applicable.
Finding new ways to straitjacket the nominal 'civilizations' or the player in the game is counter-productive. The limitations of the entire Unique system are eloquently shown by the number of 'alternate uniques' that the Mod Community has come up with for various civilizations in Civ V: if they can derive 7 different 'Russias' (including the Grand Duchy of Muscovy and Three different variations of the Soviet Union!), 3 different 'Romes', and an entire panoply of Indian' states/civilizations, then all over-riding 'unique' factors or traits are simply limiting, not enabling.
Away with both 'Traits' and 'Uniques' that define a civilization arbitrarily - give the Gamer the chance to define the civilization by the way each individual game proceeds, from the influence of the starting position through the terrain, neighbors, resources, technologies pursued, and Social Policies and religious policies adopted.