If that was the case, birthrates wouldn't be so low in Europe. Free school and healthcare doesn't support your family when rents are higher than ever and fewer people than ever can afford to buy a property to raise a family in.
Talking to my circles of highly skilled couples most cite money as the primary reason they're putting off having children.
Until you get terminal cancer and you are sent home to die. Then, I assure, a “kid” might be useful, because nobody else will give a flying fuck.
Unless one is single, there's a ton of stuff you have to worry about, from finding a job for the spouse, to learning the language, subtle to not so subtle xenophobia, navigating a different social & political system, etc. Moving with kids makes things even more complicated, to the point that we essentially only have migration from poorer to richer countries. I'd wager immigration from the west to the east is almost zero. I know some cases, but it's a tiny minority.
Then there's the fact that most people don't want to live in another country, surrounded by foreigners speaking a language they will almost certainly never speak fluently themselves.