Depends on the country. In mine women get much more time and men usually take none or just a few weeks at most. I think that's the case in most countries.
The employer doesn't pay you a dime. First twelve months you get approx. 60% of your current income from the government (BTW the same coverage people get when unemployed). You can expand those twelve months into 24, halving the amount of what you're paid. You can also apply for additional support (e.g. there's a fixed amount per kid you'll get until it reaches adulthood).
From the feedback I've gotten so far 3 years sounds bizarre to most people outside of Germany. It certainly sounds like something guaranteeing a manifestation of gender-pay-gap. But it is in my experience actually the women here, who mostly want those 3 years. And equally bizarre: how shocked many are to hear how fast mothers return to their jobs in say France or Switzerland.
Germany 1.41
France 1.98
Luxembourg 1.78
Denmark 1.74
Netherlands 1.66
Switzerland 1.44
Austria 1.38
Poland 1.27
Birth rates drop as prosperity increases and people feel less need to have children (some rejecting children entirely), and Germany is one of the most prosperous countries in the world.
Other than that you're right.
One thing I keep hearing that actually worked better in Eastern Germany were the government funded expenses to make sure, women stayed financially independent from the father (and family), if need be (as indicated by the fact that Eastern German women that experienced the GDR are usually immune to today's extreme forms of feminism).
One thing in particular was that in the GDR day-nursery coverage was at 80%, in cities at 100% and they were completely free (except for the food). FR was higher than in Western Germany during the whole time GDR existed.
Note: This is not an endorsement of high government spending or communism. The US trumps all of Europe with their FR while having assumably a far looser social security net [1]. In GDR there also wasn't much choice for where to put your kids and the methods of caring for them were certainly blind to trends like "montessori" etc.
I guess, much of FR is due to economic predictability and less to the height of income you predict.
[1] For now, since US FR is declining, against the trend in Europe. Also of interest: How much of that rate (US or EU) is inflated by migrants?
The law does not discriminate by the parent gender, so if the man decides to stay at home with the child is fine.
Including in the part of “Europe” south of Canada and north of Mexico; federally, it's up to 12 weeks of job-protected unpaid leave under FMLA, but some states have more and/or paid leave.
Men can get parental leave in European countries, so in this case your strategy is not working.