Daphne,
I understand your sympathy with the OP, given your similar situation.
However, the advice that you have given her might, unfortunately, be hampering her in acquiring the school place she so much wants.
For example, you have advised her to refuse the school that she has been offered, on the grounds that it is not a great school. That is fine in itself, if places in better schools are likely to come up in a timely manner. However, what perhaps you haven't made clear to the OP is that this is a 'game changer' in terms of everyone's roles. Up until a place is offered, the council has an obligation to find one. Now, if she turns down the place offered on your recommendation, the council is not required to do anything else, and it becomes HER sole responsibility to find education or her children. I wonder whether you made that consequence clear enough?
Also, by implying that appeals are 'difficult' and 'heart breaking' and 'stressful', you may have put the OP off what is now, realistically, the best route for her to obtain a school place. The document I referred to earlier said that something like 26% of secondary appeals in Herts are successful - which implies that if she puts a good case, and appeals to enough schools, this route may well give her a place eventually.
The school I work in (junior) is ALWAYS full. We admit children on appeal all the time - not in vast numbers, every applicant has to appeal, but enough that every class has 2 or 3 more children than originally admitted into the youngest classes.
I do think, though, that your advice to look into Essex is very sound. Do you have an understanding of why the OP has not persued this avenue actively to date? She mentioned something about 'legal issues' - are these clear and well-founded, as the OP has not always seemed completely well-infomed about the details of processes, rules and responsibilities in school admissions?