My family sharing is completely broken, can't change payment info

I've had family sharing working great since the introduction of IOS 8, but yesterday one of my kids told me that none of her apps were working any more. Both iTunes and the App store on her iPad said that family sharing setup was not complete. The link led me to the iTunes Payment screen, where I was told to correct the items highlight in red, which was only the 3 number security code. When I press Done, I'm told again to correct items in red, which there are none, however the payment type is blank with no way to input anything.

So, backwards and forwards between iPads, logout, log back in, even restored one of the kids iPads, but the same issue occurs. So searching for a solution, the internet suggested the family organiser removing the payment method and re-entering the same details. So I go into my account in iTunes on my iMac, try to edit my payment information. The last four digits of the number are there, but it's all grayed out, and there's no option to change, add or remove any payment type.

So here I am wasting my whole weekend trying to get the kid's iPad working. I can't turn off or even remove anyone from family sharing because I have two kids under 13. Their iCloud accounts are totally useless at this point.

How do I fix my payment method issue and my broken family sharing?