Low birth rates and high home values are both also highly correlated to education and women in the workforce and high income and economic growth and trendy hipster bars and many other nice things.
Picking just two items and claiming that "people don't have children because homes are expensive" is not very enlightening at all.
> As a further caveat, the correlation observed here is by no means proof that home value growth causes fertility declines. One alternative explanation could be the possibility that there is clustering into certain counties of people with careers that pay well enough for expensive homes but make it difficult to have children before 30; this could cause both trends observed in the chart above. There are many other confounding factors that could explain this relationship as well, such as the possibility that cultural values or the cost of child care varies across counties with some correlation to home values.
Moving to quieter town then you are well off and stable career-wise is certainly what a lot of people do.
I think there really is no root cause here. They're all interconnected factors.