I suspect the Brotherhood and Institute are at odds, which is ultimately what sets us against them.
Or, no matter how much the Brotherhood (specifically Lyons branch, which I think is safe to assume is "the new normal" for the Brotherhood) has changed its philosophy, the chance to acquire a survivor from before the Great War is too great a prize not to try and grab.
Third possibility, the "Brotherhood" in Fallout 4 is actually the Outcasts.