It seems clear to me that there are plenty of parallels between the Exodus story and the plight of the pre-Civil War Black population. Enslaved in a foreign land, overseen by cruel masters who "...embittered their lives with hard labor, with mortar and with bricks and with all manner of labor in the field; any labor that they made them do was with hard labor" (Exodus 1:14).
It also seems to me that this was not lost on those alive in the 1860s. The song "Go Down Moses" seems to be the most obvious connection. Even Jews were contemporaneously drawing comparisons between Moses and Lincoln after he was assassinated.
Now, I understand that "why did X not happen" is quite difficult to answer. So perhaps this is better phrased as, "If Christianity was the religion of the oppressors, why did postbellum American Blacks become Christians instead of Jews, with whom you'd think they'd've more strongly identified?"
You're absolutely correct that the Old Testament story of Jewish oppression & freedom became central to Black Christianity. After emancipation, formerly enslaved persons were more likely to remain Christians (or join free Black Christian denominations) rather than convert to Judaism. This is because
I'm speaking in broad strokes because the resulting Black Christianity (or Christianities) were extremely diverse, depending on which African spiritual practices had survived, which Christian traditions were available, and the degree to which religion featured in enslaved people's oppression or freedom. For example, Afro-creole Catholicism in eighteenth-century New Orleans looked very different from the "Africanized Christianity" that fueled Nat Turner's Rebellion in Virginia, 1831.
Undoubtedly, Christian doctrines and practices were deliberately used by slaveholders. Many "slave narratives" (written by formerly enslaved people) in the 19th century describe white preachers who preach on two texts: "Slaves, be obedient to your masters," and Paul's letter to Onesimus. (Paul tells the escaped slave to return to his master).
But enslaved people weren't just passive objects for indoctrination. They understood that Christianity was being used against them. They took key Christian symbols, stories, and practices, and crafted something new that allowed them to survive and create.
Free Blacks in the Northern states created similar parallel Christian communities. Even in "free" states before the Civil War, antiblack racism was prominent among white US Christians. African-American Christianity in the free north, like slave religion in the South, was a creative articulation of Black cultural identity. (Key moment: 1816 Philadelphia and the founding of the African Methodist Episcopal Church)
After Emancipation, Black churches were crucial social institutions for formerly enslaved people trying to create a new society. Northern Black churches were key actors in promoting Southern literacy and helping reunite families.
tl;dr: Black Christianity was separate from slaveholder Christianity, and a key site of Black identity-formation.
I'll post the second half of the answer-- about the history of American Judaism and Christianity, and Jewish-African American relationships-- after a quick videocall.