You should read the actual records of the debate rather than a heritage foundation op ed: http://memory.loc.gov/cgi-bin/ampage?collId=llcg&fileName=073/llcg073.db&recNum=13
The people debating the amendment clearly understood that it applied to children of immigrants - some opposed, some supported. But to say they never considered it had this effect is false.
I don't think anyone doubts the bolded part.
There seems to be more question if it applied, for example, to a pair of English nobles here on vacation who happened to have a kid. (tourist) Or a group of female Mexicans running with some bandits in Texas that had a kid on a raid across the border (illegal).
As you pointed out that folks would come here and live permanently and have families while not being immigrants recognized by the State and Local govs, was not really a thing to the drafters of the 14th amendment, hence the division among authorities about how it applies to that situation.
Personally I think it means what it says (much like the other amendments) and anchor babies are citizens. The fact that a new(ish) situation has evolved that the writers of the amendment didn't foresee doesn't mean we can ignore the pretty plain language. The issue with immigration and anchor babies is manufactured by us anyway. The answer is not to try to parse language on the amendment, but not to keep extra illegals here for the sake of one citizen. The minor citizen can stay, and be a ward of the state, or they can go with their parents and attempt to establish their identification and get a passport when they are 18. Then they can come and go as they please. The idea that parents and grandparents automatically get a pass to stay is the issue, not the citizenship of the anchor babies.