You are correct that Amhara elites had dominated Ethiopian politics and culture throughout the imperial period.

I wouldn't agree with your presumption that Bahir Dar "should" be the capital, given that it exists as the capital of the Amhara Region of Ethiopia. In fact, the current administrative divisions of Ethiopia only date to 1992. If you look back to the older provinces of ethiopia and compare them side-by-side to the current regions you will see the older administrative divisions don't correspond to the "ethnic homelands" of today.

I should also say that prior to the 20th century, the imperial government of Ethiopia was quite decentralized. The rulers of each of these provinces were called Negus which literally means "king", and they ideally owed their loyalty to the Negusa Negast, the "King of Kings" i.e. the emperor. In reality, there were often rebellions or attempts to overthrow the emperor by ambitious neguses.

I will also note that these kings and other nobles spoke amhara, and were "culturally amhara" even if they ruled over a province that was populated by non-amhara peoples.

The origins of Addis Ababa date to 1881, when the negus of Shewa, Sahle Maryam, founded the town and moved his capital from Ankober to Addis Ababa.

Later on, in 1889 the emperor Yohannes IV died in battle, leaving the throne to his son Mengesha Yohannes who ruled Tigray province. However, Sahle Maryam said that he had a better claim to be emperor because he could trace an all-male line back to a Solomonic emperor, while Yohannes IV and Mengesha Yohannes had a claim that went through a female line.

In the conflict that ensued, the armies of Shewa defeated the armies of Tigray, and Sahle Maryam took the throne as Emperor Menelik II in 1891 and ruled until 1913. So, as Menelik rose to power, Addis Ababa also rose in importance, from being the capital of Shewa to being the capital of all of Ethiopia.

After Menelik died, his descendants would continue to control the imperial throne through the last Ethiopian emperor Haile Selassie. At the same time, the capital remained in Addis Ababa where the House of Shewa had power.

By the time of the Ethiopian revolution in 1974, Addis Ababa had been the capital for such a long time, and the infrastructure of governance had been built up around it, the Derg and the current EPRDF government haven't chosen to move the capital anywhere else.