A few years ago, Child Benefit was called "Family Allowance". We all have a tax free allowance. (well unless you earn over c. £114K) People over 75 still get a married person's allowance.
The government manages tax in all sorts of ways to encourage/discourage activities and maximise revenue. Much of this system doesn't make sense (tax on biscuits but not cakes), and some of it seems to have some logic (tax on cigarettes).
Is it fair that somebody who earns £960 receives child benefit? Well the maximum housing benefit you can claim is £400 a week, which, even under a conservative government recognises that housing is pretty expensive, and somebody earning 960 a week only takes home £688 a week after taxes, so I think yes, it is fair.
It is also true that many SAHP's are staying at home not because they are going to the gym, but because they are caring for children who don't fit into the normal school/after school care/child minder system and therefore find it difficult to go out to work. Not everybody can benefit from the free child care provided by the education system that others take advantage of, and this isn't 'fair' either.
Personally, I believe that everybody with children should be entitled to Family Allowance, in the same way that everybody with children should be entitled to a child's library card, the NHS and schools. It's a tax allowance that recognises the additional cost of having children. However, I realise that some people might just say don't have children.