I have not read the whole thread, but worked out from the OP and on the first page that this woman is likely to be lacking in social skills, and learning disabled to some degree.
I was slightly alarmed by the lack of understanding of people in her situation - echoes of Boris J and Jacob R-M.
People who are at the bottom of the heap often do not have the confidence to drag themselves up - all the suggestions for work for her are what would happen in an ideal world, where people like her had access to family centres and Sure Start (now dead following Tories pulling the funding} to gain confidence in parenting skills and how to get into the job market.
Add in the child care problems for this woman - a challenge to sort for anyone - and her chance of gainful employment is very low, as the OP has identified.
What I would expect of society in this situation is that she should be supported, both financially and in her role as a mother, so that at some point in the future she could find her way back into work when her son is older. That would help her to be a confident mother - a job it is in all our interests that she should do well - prepare her for work with gaining appropriate skills, and give her the chance to step up gradually as she is able.
Unless you have experienced the soul-destroying situation of being bottom of the heap in this fast-paced target-driven society, it is very hard to understand what this does to people and very easy to make judgements. And to understand how the education system that we have finished up with is a one-size-fits-all system that makes pupils like her switch off and fail to see its relevance to real life.
I am sure most people would agree that a child care job is not appropriate - would you honestly want someone with a learning disability being a minder for your children? And even cleaning jobs demand a certain degree of organisational skill that this woman is very unlikely to possess: planning the day, the transport, the childcare, the route the cleaning order etc. Sadly some people are simply unable to do this. As a caring civilized society we should offer all the help we can, both to have prevented her from getting in this situation in the first place and also to support her parental role.
The role that Job Centres now have is to police the government's insensitive and rigid policies - absolutely, everyone should be in a job who is able, but so often many of those who are long-term clients of the Job Centre have pressing problems: drugs, lack of education etc., and the staff there are basically fighting a losing battle.
Better to acknowledge that fact and put the effort into helping people to be more employable and to prevent them finishing up in this situation in the first place, rather than holding the threat of destitution over their heads.