Strictly speaking the problem isn't that the kid is expensive, it is that, per your two options, 1) childcare is expensive, and 2) London / their spending habits are too expensive to support a stay-at-home parent. There are lots of other choices that one can make, although they are problem not the common choices. They might actually be happier with some of the other choices, from stories I hear of people that sat down and thought about other options.
It really has nothing to do with being able to support a stay-at-home parent or not. If you're bringing home 100k a year and quit your job to raise kids then the opportunity cost of having kids is 100k a year. The fact that you can afford to 'lose' that money doesn't change that.
Now you may think it's worth 'paying' 100k a year to gain all the non-monetary benefits of staying at home and raising kids, but that is a separate discussion.
