You know magicmummy I am not sure exactly what advice you are asking, as you are keen to disagree with most that is said here, and discount a lot of posters own experiences and thoughts.
However, I think the crux of the matter is in the fact that your girl is a popular girl, by what you are saying "She is a very confident, gregarious little girl with lots of friends, and she gets invited to plenty of parties, playdates etc, " and as such seem to have had very little experience in dealing with disappointments. This may be why she is making a big thing out of it. I do also find it odd that she is refusing to accept the situation.
On the other hand, the other girl seem perfectly happy to NOT play with your dd after school, possibly because 1. she sees no need to play with your dd after school because she has no expectations of after school playdates, and she is already very busy. Bear in mind that this little girls knows her own family and the activities they get up to, she probably knows better than your dd and you just how busy they are. And if your dd is unable to understand that there is no time, this little girl probably does.
Is your child an only child? Is she used to getting her way?
With 7 children you cannot let your kids be the boss of you, you need to stick to a routine, and I would imagine their life is run with military precision, with school pickups and afterschool activities, and it must be really hard to start throwing the routine away.
To be honest, you do seem to go on a bit, you say you have suggested many playdates, many ways of resolving the "problem".
You dont appear to be really listening to this other mum, it must be extremely embarassing for her to have to rebutt you time and again, because your dd wants to play with her dd, so you keep asking without much thought to their family set up.
It is hard for a parent to discover that not all children like your child as much as you would like, and it is hard to deal with their disappointment when they dont get to do or see who or what they want, but this is just one of the many challenges of parenting. But could it be that this family is not really that keen on your family? Maybe they find you a bit pushy?
I have two children, 8 and 5. When my oldest son was 5 and in year 1, he had a special friend in the class, and this friend was never allowed on playdates. My son accepted this, and did not keep asking. If this qualifies me to comment. 
Also, after a long day in school, my kids are more keen to see us and eachother, and go to their activities, than to play with a child they have seen and played with all day.