I'm just watching an ABC News broadcast where a Trump voter (who arrived in the US from Canada at the age of two many decades ago) has discovered that he's technically an "illegal alien" as his birth to an American father wasn't registered in the US. So leaving aside the fact that he has been refused a state pension and has inadvertently been voting illegally for 50 years, he's at risk of deportation because of policies he voted for.
This man's name is James R. Klass. The story is a bit confusing, but American journalists rarely understand US immigration rules, and for that matter British journalists don't understand UK immigration either.
Under the law at the time of his birth (
https://uscode.house.gov/statviewer.htm?volume=66&page=236), children born outside the US to one parent who is a US citizen, are automatically US citizens provided that the parent has been present in American territory for not less than ten years, at least five of which were after [the parent] attained the age of 14.
So as long as his dad grew up in the US, he would have been a US citizen from birth. He was just unable to prove it, because the first time he was asked to prove it was when he was in his 60s. This is similar to the Windrush people in the UK.
(If his dad did not grow up in the US, there is no way for him to get US citizenship except under the provisions that would apply to any foreigner. In that case, his parents should have obtained a US visa for him when they came from Canada, but they were probably not asked for anything to cross the border. Children born after 2001 to US parent(s) who haven't grown up in the US, can now get citizenship after moving to the US legally.)
According to one news article, he hired a genealogist with the help of Senator Marco Rubio, and documents showing his dad's presence in the US were found. This would mean that he is now able to prove his US citizenship. However he apparently had to go back to work to pay the genealogist, which would have been illegal if he turned out to not be a citizen.
Registering the birth is only something you need to do now, in order to get a US passport for your child so they can travel to the US. It doesn't affect whether your child is automatically a US citizen at birth or not.