I don't think replayability obliterates completely the fact to roleplay its own country, as said specifically. Not only uniques are not all replayability, far from it as I showed it (someone said 95% of it, LOL), but no uniques would also allow replayability. (do I play my country ? Do I play another one ? If it changes only city names and CLSL, that doesn't affect gameplay, I will at least be free to roleplay any country within my own ideas, not according to a gameplay scheme that may dictate me how to profit from its uniques within some game systems that are far to constitute a creditable scenery)
What I hear you saying is that you would
like to roleplay as your favorite civilization (possibly one that is not one of the default choices), but you feel prohibited from doing that. You feel that because unique attributes exist, you
must choose the one that you feel is most advantageous from a gameplay perspective, and this limits your ability to role-play.
The problem, then, seems to be a limitation not in the game, but in your approach to the problem. Earlier, you mentioned feeling the need to always take Egypt for their happiness bonuses. There are many alternatives, including taking a faith-based civ such as the Celts and using religion to generate happiness.
Very few of the unique attributes are objectively better in every circumstance. If you try, you can find a way to play and win with any given civilization. You can then use a mod, or simply rename the cities and units in-game, to match the civilization you want to play as.
If that isn't good enough, you could also download a total conversion mod that will remove unique attributes.
Frankly, I myself, and many other players, would get bored if all three civilizations were visually distinct but functionally identical. If I want to play a game with those conditions, I could play Risk, or Stratego, or Monopoly, or Catan, or chess, or any of a hundred other strategy games. Those are all great games with a fun social aspect, but I play Civilization because I could play for years and not exhaust the possible combinations of unique attributes and strategies. If all civilizations were identical, I could play maybe 5-6 games at most before it gets stale.
That is why I and many others contend that unique attributes add replayability.