I think I can understand part of what was troubling your previous girlfriend, as I was in a similar situation while dating my husband (we met in high school, our early years together were obviously embarrassingly angst-y).
OP, your current relationship may be over, but maybe I can offer some perspective if you find yourself in a similar situation in your next relationship.
My husband has a shitload of sisters and, when he was younger, a tendency to let them walk all over him.
I liked his family because they really are good people, but it was also really important to me that if we did get married, there'd be boundaries. A major source of misery with my own parents were their overbearing and rude in-laws, and I absolutely did not want that for myself. I knew in my early 20's that anyone I was thinking of marrying would be capable of sticking up for me and our potential children.
What if his alcoholic cousin drops by, asking to crash on our couch? Would he be willing to tell him no? One of his sisters needs a pretty substantial loan, will he turn her down? We have a baby and his mom becomes really controlling and imposing, would he be able to talk her down?
I was becoming really concerned, because after a couple of years it was becoming more and more apparent that my then-boyfriend wouldn't be able to do any of those things.
He would cancel dates at the last minute because his sisters would either take his car or leave him home alone to babysit their younger siblings.
His sisters would take his gift cards and coupons and use them for themselves.
Cell phone, Facebook, email, personal journals, they'd rummage through everything while giggling (even in front me).
And I think what really worried me was he didn't think they were doing anything wrong, at all.
I was kind of devastated because I was becoming more and more convinced I would have to break up with a guy I found absolutely perfect in every other way, but to be sure, I remember picking fights over really, REALLY stupid shit to test how he'd react.
It was pretty messed up in hindsight, but after talking to a handful of other younger married friends, I've found that a lot of women do similar: testing to see how a man would respond when forced to side with his significant other or his parents/siblings, because NO ONE wants to feel trapped with overbearing in-laws.
I don't think your girlfriend was really upset about the Facebook photo (a little jealous, yes, probably). I think she wanted to know you'd back her up over your sisters and she escalated it when you didn't.
She was probably incredibly excited about the drive home with you from the airport, and then equally disappointed that you decided to share that moment with your sister, someone you knew she wasn't comfortable with.
Revoking an invite to a family function (a serious girlfriend practically IS family as she is a potential wife) is pretty insulting. I have never been to a family event where a long-time girlfriend isn't expected to attend.
OP, I'm not trying to make you feel any shittier about your breakup than you already do, but I can easily see you having similar problems in future relationships. It's rough, but part of being in a healthy, adult relationship is figuring how to balance your dedication to the partner you want to share your life with and the family that grew up with you. It is very possible to be an awesome son and brother and become a very difficult husband/father.