@Livingincanadaafter19yearsinlondon, I don't think you have misunderstood, I think we just disagree.
In the UK people are taxed separately, not as couples or households. It is not possible for one partner to pay the tax of another. If the hypothetical wife worked, she would be taxed in her own right, which in the situation you describe she is not. Every parent ferries children about, makes lunches and so on, and a lot of voluntary work is also done by people in work passing on professional experience or doing things like running Brownies in the evenings. It is untrue to suggest that SAHMs provide all the voluntary work or that working mums don't look after their children. Of course they do, and it's quite insulting to suggest otherwise. The only difference for a lot of parents is that a SAHM has her children with her for an hour or two longer when they finish school. Otherwise, early mornings, evenings and weekends are the same whether parents work or not.
I feel no-one should be left destitute in older age, but I am not convinced that pensions should automatically go to anyone who hasn't supported themselves if they were capable of doing so, whatever the reason for their choice. Means-tested PC should be the only way those who chose not to work get a pension, which is what I meant when I said a pension should not be a reward for reaching a particular age. They are supposed to be a reward for working and contributing to society, not just looking after your own family. I think it is different when children are babies, but I wouldn't pay contributions to parents of children of school age. If a woman has a high-earning spouse she would fail the means-test but would, presumably, be financially ok, as she was when younger. I don't think the taxes of everyone else should be redistributed to pay for her to stay at home.
I wasn't describing the Canadian system in my last pp above. What I mean is that people pay in depending on income, and take out a flat rate. This means that higher earners subsidise lower ones, but higher earners also get higher tax relief on pension contributions in the UK, and pay a lower rate of NI on earnings over about £50k, so proportionately they pay less, freeing up more money to invest in occupational or personal pensions outside of the state system.