Neutrality is a cherished concept in mediation, but the term is interpreted differently by different mediators. Some take it to mean being strictly impartial and fair toward both sides in a dispute, in the way that a judge should not let his or her biases affect the outcome, or show favoritism toward one side or the other. I take it to mean being indifferent toward the outcome of the process, leaving the parties to achieve a resolution acceptable to each side, but assisting both sides in satisfying their interests to the greatest extent possible. To do that mediators play a different role than judges or arbitrators, at times coaching one side or the other on their negotiating tactics, and at other times playing the devil's advocate. It's more of a shifting partiality, than an attempt to remain impartial.
That is why the ideal mediator does not necessarily resemble the image of blind justice, holding a scale that awaits the evidence to be piled up on each side. The ideal mediator is more likely to be someone who each side thinks the other side will listen to. In some cases, that ideal mediator might resemble a judge or other authority figure, if each side thinks the other is prone to respect such authority figures. In other cases, it might be best to bring in a friend of the other side, or someone with similar background and sympathies.
Michele Fiore played a critical role in talking some of the protesters into surrendering to federal authority. Ms. Fiore is not someone one would think of as a "neutral" party in the dispute, being famous for posing with guns and taking outspoken conservative positions. But she was someone the protesters viewed as sympathetic to their cause, someone they would listen to, and someone they felt they could trust. She was one of them. Her role was to assure the protesters that they could trust the government enough to lay down their weapons and allow themselves to be taken into custody, and that in return they would not be harmed.
Certainly the FBI must have used a lot of tactics well known to negotiators and mediators in bringing about a peaceful resolution of this dispute. But when it came to finding someone who could act as a "neutral" between the government and the protesters, Michele Fiore fit the bill well.