As a European, I think that this problem has many root causes, not just one.
There are always exceptions, but as a rule, I do not believe that it has really anything to do with financials, most European countries now do support families with children financially - I think it is way more than what is offered in the U.S.
Imho, the main factor is that individual fulfillment is the most important thing today for any average European.
The job, hobbies, friends, travelling are all much more attractive than getting up at 3 am because the little one does not stop crying. People have the choice not to get children and they at least delay it as much as possible, ending up with only one or two at most.
The expected ideal is to schedule everything perfectly, i.e. the job, the fitness club, yoga, going to Paris over the weekend, ski holidays, going out with friends four days a week and so on.
Children make all of this so much harder and less fun.
They cry all the time, they get sick when you want to go skiing, they throw up on the new carpet, they will not let you watch your favorite TV show in the evening because they would rather play with dad.
So in short: Children are a nuisance. You try to avoid nuisances.