The clause was limited to prevent, for example, the children of diplomats from receiving automatic citizenship. In fact, the 14th Amendment was determined to not apply to members of Native American tribes in Elk vs. Wilkins, because their primary allegiance was to their indigenous nation. They were later granted birthright citizenship by The Indian Citizenship Act.
The actual text: 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.
If Trump's Administration challenges birthright citizenship, it will be based on the claim that the children of illegal immigrants owe their primary allegiance to their parents country of birth, not the US, so they do not fill the intention of "subject to the jurisdiction thereof". This may have other consequences if they win on this basis, though. Can we jail people we are saying are not subject to our jurisdiction? IANAL.
Indeed.