There are places in the world with far more horrible economies than Greece that have no fertility rate problems. There are places elsewhere that have much better economies, and similar fertility rate problems. It's not economics, not even a little.
When you teach your youngest generation to not want to have children, not only do they take the lesson to heart, they teach it to whatever children they do end up having by accident or apathy. Instead of going away or even just lingering, it is self-reinforcing.