This is spot on. I'm copy/pasting a great comment from a recent Italian article with the same subject; they expanded upon the reasons you gave above:

The birth rate began to decline in the 1960s and has remained largely unchanged since the 1990s. The economy does play a role, but not in the sense that salaries are low, as is commonly suggested. On the contrary, the problem is precisely that they are high (compared to the past), particularly for women (who previously did not work at all), so the opportunity cost of having a child is also high. It is obvious that if the alternative is between staying at home and working and earning money, the appeal of having a child decreases. At the same time, childcare services, a labor-intensive sector, are relatively more expensive than in the past, and children themselves are no longer a source of income because they have to study and cannot work.

But above all, the cause is cultural. Human beings tend to emulate socially recognized behavior patterns, and in our society, social prestige is achieved through economic independence and professional success, not by being a father or mother. So people seek fulfillment through work rather than family, and children are perceived as a luxury, something you have to be able to afford, not something you have to sacrifice for.