The op was alone and the others were in a group, so I think the onus is on the group to initiate contact.
It's a nice idea, but in fact there isn't any onus on anyone to initiate contact with anyone really - if the person on their own wants friends but the people in the group aren't particularly looking for new people to hang out with, then the onus is on the person who is trying to initiate contact.
I think of the school gates as like a commute - I used to get on the train with the same 50 or so people every day. All we had in common was this train journey. I occasionally smiled or said hello to the people that I saw at the station, and sometimes nodded or smiled or even exchanged small talk with one of the regulars from the train if something particularly exciting happened, like a delay or a ticket inspector.
Some of the people getting the train knew each other, either from working together or from previous train encounters, so they would sometimes sit together or move from their seats to talk to one another as other people got on and off. I never expected anyone to include me in their friendship groups, just as I wouldn't have invited a lone traveller to sit with me and friends on the days when I would chat with colleagues or other commuters that I knew.
If someone had wanted to chat to me and was looking for a train buddy, I would probably have shared a bit of mindless chat for the ten minutes that we were on the train, but it would have taken a lot for me to want to meet up with them outwith our usual interaction on the train itself, whereas I would occasionally meet up with people that I knew better for lunch or a drink after work, or even a coffee in the station cafe if there were delays.
School is just the same - you spend a certain time in the same place with a range of other people for reasons which are nothing to do with your personality. Wanting to be included in someone's existing friendship group is understandable, but expecting to be automatically included just by virtue of your sudden presence at the ritual of picking up/dropping off is a little unrealistic.
OP - I'm sorry that this experience affected you so much, and I hope that it was a case of mismatched expectations rather than anything more sinister. Could you volunteer for something at the school so that you have more opportunities to meet other parents outside of their existing friendships with one another?