I found this tough reading, for a couple fo reasons. I also have a runner, we had to get police involved late last year when he ran off at the school gates and was nowhere to be seen. Thankfully he was found safe and well, but it was terrifying. I also have two year old and other children to consider in these circumstances, so very similar situation as desrcibed by OP.
Clearly mum is having issues and struggling, in my opinion she's crossed a line with the dragging and heated language. However it is not the OP's responsibility to offer help, a kindness yes, but it could also cause further agro, especially if you are not known to the woman or her kids. Please don't feel like you could have or should have helped, strangers are often forthcoming with me, and while that is nice, the passing nature of such interactions means that it's ultimately quite futile...what is needed is a solid support network working between the home, school and any other agencies involved.
Contacting school rather than direct to SS would be my advice, like the mum said they are already aware. Mum needs to be discussing possible transport assistance or possibly a gate greeter to avoid these problems. Funding might be an issue, but they won't know til they ask. It would probably also be of huge benefit to the mum if she was able to get some guidance on how to properly and safely restrain and guide her daughter, to reduce risk of injury to both parent and child.
As for the two year old, was it in a pushchair? Whether my little one wants to or not, he has to stay in the pram while his brother is with us, simply because I need to be able to safely park him somewhere should the other run. I also have a buggy board and other mobility helpers (everything from a lightweight scooter, to tethers). DS is a bit younger than the child mentioned, but there should be plenty or accessories for an older girl to help her along, and if her history is as bad as the mum claims, she should qualify for low rate mobility support. Perhaps mention these suggestion to the school if you're talking to them?