"As I said earlier, they did care for one another at one point in time, so why is it so hard for them to be able to muster that up enough to make this work?"
To answer that question, I think you'd have to dig to the root of what made them decide to break up in the first place. That would probably take some heavy-duty communication on both their parts, and may require a (poly-friendly) counselor. Did they both decide to break up, or was that mainly his idea?
"Yes, he was the one to bring up the whole *possibility* of the poly thing initially. Then we sat down and discussed it and we both agreed to it, then he broached it with her when she still lived across country, and she agreed to it as a possibility and moved down here. We completely funded her move."
The desire to be with one's kids is a powerful force. It is unfortunately possible that he initiated all this because it was the only way he could think of to get his kids back, even if he didn't *necessarily*
intend to snatch custody away from her. The point is, getting her to move back to your part of the country does give him considerable access to his kids that he otherwise wouldn't have. So his motives have to be questioned. It would unfortunately explain why he doesn't *really*
want to have sex with her.
I get the impression that she wants to repair her relationship with him, but he isn't quite so interested in repairing his relationship with her. It sounds like he's doing the bare minimum in that area, of what's necessary to keep his kids around.
The situation might not be hopeless though. Even if his motives aren't 100% pure, there's a chance that your encouragement to keep him and her communicating may lead to an insight of something that could fix their relationship. But it's also possible that this is a job for a professional, so if you can find a poly-friendly counselor (for all three of you), by all means do so.
At the moment, what I see is that his current amount of effort is just enough to keep her -- and their kids -- around. Plus he now has a back-up plan to take the kids from her if she tries to move across the country again. Which adds up to less incentive for him to try to improve things with her any more than he already has. So you have your work cut out for you if you want to fix the him-her dynamic. There's a limit to what you can do since you aren't in their dyad. Both of them need to be willing to put considerable effort into this.
What might make this seem the most worth it to him? perhaps that it would be good for his kids to see him and their mother getting along better? that it would reduce the hassle required to keep his kids in his home? If a selfish motive is what he has to start with, maybe it can morph into something more sincere after his relationship with her begins to improve. But once again, I think you'd need (with lots of their help) to dig down to the reason/s he and she broke up in the first place.
Sorry to be the bearer of bad news. We can't be mind readers, so we can't know what's going on in his mind, but we can make guesses based on the known facts. I guess the question now is, do you still feel like it's worth it to you? If so, prepare yourself to work with what you have, even if what you have isn't as ideal as you originally believed.
I'm pulling for you to discover some good news.