Bookworm, while I appreciate the points you have made, you seem to be missing the fact that in this situation, one of the parents has not put the interest of their kids first to start off with, and neither has the OW. The fact that you would put your own needs before that of a child means that the person left behind is naturally going to have doubts about the standards you have set.
My scumbag Ex and myself worked hard to raise our kids with a certain set of values that included loyalty, honesty and respect for others, things that most parents would value in their kids, and the way these 2 people have behaved doesn't reinforce that, so naturally there are going to be doubts about a woman who places herself between a father and his kids reinforcing values that we held dear.
Any decent parent is going to have concerns putting their child in any situation that is going to cause them to be conflicted, or confused, or uncomfortable and cause them anxiety, it doesn't matter how you dress it up, these people have not acted like good people, regardless of their normal personality strengths, and when it comes down to it, if a parent has concerns about their childs welfare they have the right to voice it.
As for introducing another partner on the other side, it is not a double standard. When a new partner is introduced to a child under the right circumstances, and with the best interests of the child/ren in question and in a respectful manner there is never going to be an issue regardless of the circumstances. My personal issues with my situation will be put to the side so that my son won't feel like he is being disloyal or picking sides, and that will only be guided by their behaviour from now on in. At least that is my intention, can't say as it will actually go that way. Bitterness and assumptions are usually fuelled by assholes. I might be wrong, but judging from your responses I get the impression that we are viewing the situation from 2 different viewpoints.