Maybe not Asia-centric, but I can name 11 civs whose capitals (and most of their territory) are in Asia:
Japan
China
Korea
Mongolia
Siam
India
Babylonians
Ottomans
Persians
Arabs
Byzantines
And another 4 more whose territory has spilled over into Asia:
Russia
Greeks
Romans
Huns
I might even be missing some more (Egypt perhaps?). I'm not complaining, mind you. As far as civs go, I say, "the more, the merrier."
Okay, let's start with definitions. What are we calling Asia, and what is Asian centric? Well, if you are lumping everything that is part of Asia as "Asia" in this case, ignoring any definitions of the Middle East (which generally includes parts of North Africa) and blurring the lines of Europe and Asia (which are poorly defined anyhow as they are the same continent), then yes, the following can be considered "Asian":
Japan
China
Korea
Mongolia
Siam
India
Babylonians
Ottomans - Depends on definition
Persians
Arabs
Byzantines
As well as, depending on definition, the Huns. The following however cannot:
Russia
Greeks
Romans
The Greeks are most certainly European, regardless of what territory they held for what time in Asia. The same can be said of Rome and Russia. The reason being is that by the same loose definition you could define England as North American, African, Asian, Oceanian, South American and pretty much everything else, Rome would also be African as well... But I guess you didn't directly include them, so that's something.
Now, if you are to define Asia as the entirety of the continent that is Asia, suddenly things become very different. Asia to many people refers to East Asia and less commonly everything in the continent Asia that is not considered Middle Eastern. If you do define Asia as the whole continent, and there is nothing wrong with that, then the list you presented is a pathetically short list barely covering the vast number of Civilizations that has covered the largest continent that people has inhabited. Considering the shear number of people, the Empires and the fact that it was the birthplace of what would become Western Civilization, only having 11 in comparison to 14 for Europe, effectively the baby brother of what you have defined as Asia. So no, even with that, there is nothing "Asia-centric" about the game still, maybe even light on all things considered.
Personally I rather break the lists up into better bins than just that though. So let's try a different set of definitions:
Europe:
1. Austria
2. Byzantium
3. Celts
4. Denmark
5. England
6. France
7. Germany
8. Greece
9. Huns
10. Netherlands
11. Rome
12. Russia
13. Spain
14. Sweden
Middle East:
1. Arabia
2. Babylon
3. Ottomans
4. Persia
North Africa:
1. Carthage
2. Egypt
Sub Saharan Africa:
1. Ethiopia
2. Songhai
North America (excluding central America):
1. America (United States)
2. Iroquois
South and Central America:
1. Aztecs
2. Incans
3. Mayans
The subcontinent:
1. India
East Asia:
1. China
2. Japan
3. Korea
4. Mongolia
5. Siam
Pacific:
1. Polynesia
Of these, the ones that stick out as under represented are:
1. Sub Saharan Africa
2. East Asia
3. Middle East
This is based on the size of the regions and the historical civilizations that are currently overlooked.