FraidyCat, there is a point in there but you've over-egged your pudding so much you've drowned it.
In this country children have to be looked after by someone, by being taken into care if necessary. Making a point that relies on us accepting child neglect and endangerment as normal, or as one kind of reality, was not going to fly.
So, a child has to be cared for but that could be through paid childcare, relatives, older siblings helping out, parents working different hours, or parents working while looking after the child. Lots of people do live with these sorts of arrangements and many manage to pay nothing for childcare by juggling their work.
For many families one wage would not be enough but the children are still cared for. It is true to say they need to be cared for, somehow, in order that either parent can work - that is just indisputible fact, there's nothing 'nonsense' about it.
But, yes, having a parent devote their time to caring for a child and not doing any paid work, is a luxury. It does rely on the other parent having a reliable income (or considerable inherited wealth). It is essentially a middle-class phenomenon (in the sense that the one wage has to be a reasonable one). Often exaggeratedly so because the second 'sacrificed' wage was a good one too, more than enough to pay for childcare. (Though also often not, especially with more than one child).
So, you may have a point that a FT SAHP has made a choice (jointly with their partner) and is enjoying the luxury of being able to devote all their 'working' time and energy to their children. The counter-example though, is a parent who has to work at night and juggle shifts with their partner, who they may hardly see, or someone who childminds or does other work compatible with childcare.
Though, you seem to be describing a single parent, who needs to work to feed their child as no-one else is contributing. That's where the safety net of benefits comes in if they cannot work but is beside your point about childcare being needed to enable work.
Thus, 'needing someone to care for your child so you can work' is a universal truth in the UK but does not contradict your idea that work is a necessity if you're poor.