“...immigrants were welcome and no one worried about their children becoming citizens.”
You need to do some checking up on American history. Americans were historically very careful about immigration. That is why there was no immigration between about 1920 and 1965. The time when immigration was at a high-water mark only involved a few million, and they had to have papers.
Many were sent back.
Immigrants in the early 20th century had to also have a sponsor who would be financially responsible for them.
There was no welfare, housing, medical, food stamps-etc.
The name and address of the sponsor was right on the ship’s manifest.
We're talking about 1862. We had just passed the Homestead Act and were happy to have immigrants come to settle the West. There wasn't any such thing as illegal immigration except for slaves who were smuggled in.
We had over 25 million immigrants during the "Great Wave" from 1880 to 1924. The first quotas weren't passed until 1921.
It was a different time in our history and you can't project today's problems onto the circumstances of drafting the 14th.