It's strange for you because you're looking at it too abstractly. Real people is a bit more complicated. One thing is clear though: there are never easy answers in life; every decision costs us something.
Please don't patronize me. I am looking at this like you do, just with different priorities. You want everyone to walk away happy. I don't want someone who is not at fault to have more trouble than she already has (a damaged car).

You're assuming that (A) was totally wrong and (B) totally right but things weren't that way. Both tried to blackmail the other in one way or another. And in my opinion, it's perfectly normal that people treats their perceived enemies that way.
No, immature people involved in an accident treat each other that way. But in my mind, the best advice is to figure out who was at fault. You were able to assist in that. I would say: "A, it was your fault and I can confirm that. You have insurance, so it's not a big deal. B should not have to pay anything, I am sorry." And if A hit B, B was right.

(A) did not do it for the money; she wasn't going to lose much anyway: she had full insurance. She did what she did for the same reason you're telling me that you would prefer to pay the fine than to give money to the other party: a sense of dignity. It's the same action, but a different motivation. It's not some evil act as you think; it's just that (A) genuinely believed that she did not hit.
You did not initially say that she believed that. Arguing over who is at fault has nothing to do with blackmail.

It's impossible to reach an agreement between parties if they are not willing to move from their position. What I did is to suggest (B) a way to show (A) a good disposition by offering her to pay a part of the deductible. This had a positive effect in (A) because, as soon as she realized that (B) didn't want to simply get out of the problem but to cooperate to solve it (remember that A believed B hit), she became more receptive to the idea that (B) didn't hit. After a few minutes of conversation and explaining her what I saw, (A) admitted that she hit.
And after she admitted that she hit, she was still willing to take B's money in the solution you proposed. She has no right to this money. You were able to explain to her that she was at fault, so what made you and her think she was still entitled to that money? You are both ripping off B. And this refers to the time AFTER she realized she was hit. At that point, taking money from B is ripping her off. You knew all along that A was at fault and suggested B to pay because you felt it would help resolve the situation. And I say that B should not have to pay for anything.

The ultimate purpose in anything is to have a positive effect in people, not to follow some stupid set of arbitrary rules.
Ok, so I drive along and someone does not pay attention and hits me. My expectation for the person to assume full responsibility is an arbitrary rule? The person denies that s/he hit me and I have to offer to pay for something that was not my fault to get her to think? No, I don't think so. This rule is not arbitrary, it makes people repair the damage they have done because some obviously are not willing to do so.