Decline happens also in territories that had been urbanized decades or even centuries ago but had positive fertility rate until 2010s. As I said you can pull a patchy blanket of micro-theories explaining each region but not one theory that accounts for them all.
Generally when people say urbanisation is the cause of fertility decline they mean people moving out of 7 child families at subsistence farms and rice paddies to city factories. Not any developments in Denmark or the Netherlands in last 150 years.
Large cities are inherently inimical to living in large families. And yes, it was apparently the case even in the Roman Empire.