The weird part is that the author seems to take pains to draw a distinction between reason and rhetoric, but it seems her general conclusions conflate the two. If I have a conclusion based on logic, and someone comes at me trying to convince me with rhetoric based on ethos or pathos, it's not going to be convincing to me no matter how many points it scores on the audience. But that doesn't mean logic won't change my mind.
And most conclusions along the lines the author describes are normative conclusions. If, through logical debate, it's discovered that both parties are reasoning correctly, basing their conclusions on different moral (axiomatic) values, and neither has changed their mind, that is perfectly okay and should be considered a victory for responsible dialectic. Dialectic is about finding shared understanding and unsound reasoning, not about humiliating your opponent for having different values than you.
I haven't taken a close look at Kialo but if, for all their structure, there isn't much effort spent on the logical relations between statements, then... well, it might still be helpful to a degree, but I can also see it inviting a lot casuistry.