Sorry, long post coming:
Church of Christ member weighing in here. (Not to be confused with the United Church of Christ.) To us, church history is the beginning of The Way, which is what first century Christians called their movement before the name "Christian" was first used at Antioch.
The study of the Acts and Epistles are the main foci of this history, to discern what types of issues the early Christians faced as they tried to follow Christ's teachings. There is no unified central authority in our Church, as each community of believers in the New Testament only shared their ideas with one another but did not have someone apart from Christ who was the "last word" on the way congregations conducted themselves. We do not use intrumental music because there is no evidence of it used in worship in the first centruy. Does this mean we think people who sing worship songs with a piano are going to hell? Of course not. How absurd. But we simply to choose to be as connected with our origins as possible. So, and hold onto your socks for this one, as a woman I do not lead songs, prayer, or sermons in our worship.
Traditions that came after the time of the New Testament are not usually studied in our church, since the basis for the Church of Christ was to return to the ways of the first century Christian worship and doctrine.
There is much study of the various extant manuscripts in the Greek, as well as the differing translations, the validity of each, and the need for careful study of etymology in our discernment. For example, the word church in the aramaic, (ecclesia) simply means "called out of." That is not the way most folks think of it, but it helps to know this when reading scripture.
We seem to delve a lot more into history in our study of the Old Testament, with the founding of the Jewish nation, the changing dynamics from judges to kings in the political realm, the struggle of the people to keep God's commandments, the prophesies of the coming Messiah, the Jewish codes of law, and the Jewish interpretation of that Messiah as one who would rule an earthly kingdom.
What is not studied is the solidification of the later church that became the captial "C" church of the RCC.
So, yes, we study church history, but not the kind I think the OP was referring to.