It wouldn't have happened. You said she was a single parent.
Let's assume, and I know this is just an assumption and a massive one at that, that she was on benefits or a low income (assumed because she's in council housing, a single parent of young children, and has been granted a larger accommodation meaning the council still assume responsibility for her). Budgeting was a primary concern for my clients. I worked with people who were homeless or insecurely housed. I spent hours with them. If it was, let's just say, a £50 fine, that can be a huge amount of the person's weekly income or benefits. As an example, that's over 65% of a weekly amount of basic benefits such as JSA. I don't know the % of a low income in general or her income in particular. When the client then doesn't pay or can't pay then the council must make a decision, punish the person by adding interest to the fine or dismiss the fine. Neither helps.
I love budgeting. I love finances. I think I'm good at them. But someone already in the position of being on a low income and being punished with a fine rarely worked in my experience. It was my job to get them to attempt to deal with it - usually with an agreement of a small amount each week over a large time to write off the debt - often around £1 / week until cleared. Try telling a client it'll take 12 months to clear a fine! "Might as well just not pay it, what are they going to do about it anyhow!" Often those agreements were broken and eventually I'd seek to get them written off - costing the council time and money to deal with the initial billing, chasing up of fines, and eventual write-off. Most of my clients weren't inspired to fix the problem when presented with a fine, they'd just get angry. Who grassed them up? Why do they have to pay? It isn't their fault that...!!!! You already posted her letter. She has a young child and can't get to the bins. Her excuse to the council when presented with a fly-tipping bill is unlikely to differ no matter that she passes the bins daily.
I'm not saying that she's like my clients were necessarily. Just trying to explain why a fine isn't a simple thing and why it doesn't always work. None of this means I don't side with you. I don't think you should have to live with sh*t outside your door. I completely think that something needed to be done. My solution is long-winded, expensive for councils and includes lots of support and assistance measures.
I'm glad for you that she's moving and hope that you get brilliant neighbours this next time around!