Actually its not been part of our legal tradition for "over a hundred years", only since about 1965. The tradition before that required some legal standing of the parents, such as residence, for citizenship to be acquired. The story at the beginning of this thread provides the facts regarding the original intent and meaning of the 14th Amendment. It has, like many laws, been selectively enforced. AAMF, the "privileges and immunities" clause of the 14th. Amendment is continuously ignored by federal courts to the detriment of US citizens. So misapplying one part of the 14th and ignoring other parts is OK with you? Might makes right, is that it? The power of the state is all powerful!
The problems of illegal won't go away because we wish it so. It is expensive to all, even you. The 14th. Amendment is flawed by its disuse. It will be remedied someday to support legal immigration and refuse residence and citizenship to those who are here illegally, or there won't be a US to be "a light of freedom for the world".
I think you're wrong. And I an article that says we were denying citizenship to any "illegal" babies before 1965 is also wrong. I dare anyone to find a pre 1965 example of anyone who could prove they were born in the US being denied citizenship because their parents were "illegal."