I understand why the LW feels the way she does, but she needs to shift her perspective from herself and her children's immediate needs. In a way her reactions are a self-centered as she says her brother's are. Her brother sounds completely freaked out and unnerved by the children's problems, unable to cope. She already knew that when she contacted him and asked him to be a guardian for the children, which... they're "financially well-off, plan to have kids, and are at heart the sort of people [she'd] expect to rise to the occasion" does not mean they have the emotional capacity to take care of her kids, and they have also indicated very clearly that they can't handle her kids, at least not right now. Some people can't. Some people do not have the capacity. It sounds like her brother is staying away, and 1, that's a message in itself, and 2, it's better for everyone if he does, if his aversion is so strong.

There may be other things her brother could do in the event of her death (for example, be a trustee for funds managed on the children's behalf) which would be best kept separate from day-to-day care. A solid trustee is hard to find, and she shouldn't burn bridges with her brother because he has not pivoted to make her children the center of his life.

Lastly, his feelings could change. That isn't her job or anyone else's, but not pushing his boundaries and giving him space is respectful.