Well, if newborns were treated the same as immigrants are now, that's what it would be, wouldn't it?
What, are you under the impression that the state tells immigrants whether or not they can exist?
Citizenship is about rights, not existence. An immigrant who is currently in your country without citizenship still exists, and is still in your country - but he cannot vote, cannot access benefits, cannot lawfully work... there are lots of things he can only do if he becomes a citizen.
However that kind of society would be largely theoretical. Societies that control their borders and have immigration controls are a reality.
When reality is unjust, immoral, or unpleasant, it is our duty to call for change.
Some things you can change, some things you can't.
And anything that exists only as a result of legislation, you can.
Obviously.
There is huge societal momentum behind procreation and childbearing being in the domain of the parents rather than the state and it's not likely to change anytime soon.
Or ever; but as it is completely irrelevant to anything being discussed here, it really doesn't matter.
Immigration on the other hand is a problem that can be solved here and now. Besides if we were building utopian societies, the solution to the refugee problem would be to fix Iraq and Syria and every other country in the world so that people would have equal rights wherever they are, not only if they manage to smuggle themselves to Europe.
If you had a fictional nation - let's call it 'The Confederate States of America' - in which some people were full citizens, with the vote, access to government services, access to any job for which they were qualified and to the educational resources to acquire those qualifications, etc.; and other people were not full citizens, and had no vote, no rights, and were not even allowed to go to otherwise public places without permission from the full citizens - That would be a desperately unfair society, and one that needed to change.
Of course, the ruling elite all have the vote, so they are a democracy; and the rules that render the majority of people non-citizens are perfectly OK, because they were legislated by elected representatives of the voters.
If a bunch of non-citizens decided that they wanted to move from the squalid place they currently live, and to go to a place where there were good, well paying jobs, the ruling class would, of course, entirely democratically, ban them from taking those jobs; vote to build a fence to prevent them moving; and vote to catch and return any non-citizen who managed to get past the fence.
Now obviously such a situation would be abhorrent; and the good people of the world would argue that the limitations on these people should be removed; they should be free to go where they choose, and should have the vote, and be able to take jobs wherever they can find them.
Quite why this is a problem when it happens in the Confederated States of America, but not a problem when it happens on planet Earth, I do not know.
Rigid national borders are not a law of nature - they are fairly novel, having only really existed for a century or so. But in that short time, they have led to a massive imbalance between where people are, and where they want to be.
It is time we stopped pretending that we are democratic when we allow rich westerners to vote for laws restricting the movement of poor people who have no say in the matter.
It is time we dismantled the rigid borders around nations, and let people go where they please.
It would be trivially easy, in the information age, to restrict welfare such that people cannot move for the sole purpose of gaining access to it. That seems to be the only legitimate reason for tightly controlled borders; simply setting a minimum residency period to qualify for benefits ought to serve the same purpose.