One problem here is that when the 14th was ratified there were no illegal aliens.
There were no restrictions on immigration at all at the time, and no really significant ones till the 1880s and 90s. Some states had restrictions but they were declared unconstitutional in 1875.
Since there were no illegal aliens, the issue was obviously not considered by those who wrote the amendment.
Just because you want all the minutia included in a law does not mean you get it. "All persons born or naturalized in the United States, and subject to the jurisdiction thereof, are citizens of the United States and of the State wherein they reside," is pretty specific, if you ask me. Other law already defines who gets to be a citizen and just being born is not a definition. At least one of your parents has to be a citizen already. I know, my father was one but my mother was not until I was three years old.
Those in this country illegally are not citizens, period. Therefor they cannot pass on a US citizenship. The children born of illegals are illegal themselves and are the problem of their parents, not the taxpayers.