Or what if A always caused B, but the impact was slightly less than if B occurred without A? In that case, the sign on A might be negative, but its presence would actually tend to increase the probability of class=1, it's just that the positive impact has already been counted by variable B.

Maybe you try to avoid this situation by adding in an explicit interaction term of A*B, but then how do you interpret the impact of A since you now have more than one coefficient?

If you feel confident making assertive statements about what has been learned by looking at an equation fit on multi-correlated data, then your mathematical intuition is much stronger than mine!