I've just looked at the pictures and that post is rotten to buggery, surely you can't be liable. And as for being an expensive fence from what I can see you're looking at what, new post and maybe rails, the hardest part would be digging the old post out if the concrete is rock solid, if there is any concrete. An apprentice fencer could do that in his lunch hour, while texting, and eating lunch. It's madness to even think that an insurance claim is the way to go. I'm waiting to see what the insurance man says.
She's got to be pulling a fast one or just really ignorant