I doubt this explains the world-wide phenomenon, but regionally sure. I remember in the 90s when studies brought the Nigerian population estimates down this triggered a drop of growth forecasts across sub-Saharan Africa.
Edit: changed world-wife (which sounds interesting demographically) to world-wide
Sure, it is quite far-fetched. However it is extremely uncommon that we experience unified social trends all across the board, from liberal Finland or Japan to North Korea and Taliban-run Afghanistan. Usually there are odd reversals and exceptions here and there; not this time apparently. And we still lack a satisfying theory that could account for fertility decline in every country.
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.
Edit: changed world-wife (which sounds interesting demographically) to world-wide