I don't think anyone would have any problems with that pov. My point is though, that from the outside it's very difficult to see who is perfectly capable of earning their own living and who is not. With LP's I think when the family split happened is probably one of the most important factors. A LP who split with her child's father when her kids were 5 and 2, and they are now 9 and 6, may be perfectly capable of going out to work because as a family, they've all settled down to the way things are now, childcare is sorted, school is sorted, their lives are stable etc. Whereas one who has split when the kids are 12 and 15, may actually need some time out of the labour market to negotiate the very tricky waters of traumatised teens and the real dangers of taking your eye off the ball when dealing with them. So from the outside, the first family may look like the mother might find working more difficult, while the second family looks like she's perfectly capable of doing a paid job because the kids are older, when in fact the needs of the two families are exactly the opposite.
Re your £30 example, I think that comes down to the unwieldiness of bureacracies - I'm a bit surprised that the absent parent's income isn't taken into account. Couldn't agree more that it is a ridiculous situation, but tbh I expect government bureaucracy to be mad.
And yes of course you're right that if a family split happens, income will plummet. At least, the income of the person with care and control of the kids will plummet. Although I do remember reading somewhere that 5 years after a divorce, men are better off than they were when they were married and women are poorer. Hmm...