I dont think that the 4th path will be a Black Eagles branch specifically; after portraying every house as equal to the others. This would favor the Black Eagles. I think that every House will have Path A and Path B will be common for all three houses. Path B will be a unification path. How many of us, given the chance, would spare opponents and lords of the other Houses?
The game is titled "Fire Emblem: Three Houses", not "Fire Emblem: Black Eagles and the other two"