Among many problems with Trump's citizenship executive order (such as its obvious conflict with the constitution) is a practical one: birth certificates, which are currently the primary proof of citizenship for those born in the US, are issued by state and local authorities. These agencies do not (and aren't equipped to) determine or record the immigration status of the parents.

This means that citizenship would no longer be established with a US birth certificate or any other primary document.

In practice, birth certificates would likely be accepted MOST of the time as establishing citizenship, but those listing parents with "foreign sounding" names would be subject to open-ended scrutiny and investigation. And years later, people born here whose parents are dead or unavailable might have no way to definitively prove that they're citizens.
One of the issues is that for birthright citizenship (as opposed to naturalization), one is not "granted" citizenship through a formal process or application, but rather simply has it from the moment of birth. The first time a person born here has to interact with the federal government to prove citizenship might be years later (e.g., applying for a passport), if at all.

@mattblaze I know a couple people who have had problems with that. They're absolutely citizens, but they have to prove it.

One is even harder - they're citizen-by-adoption-as-a-child. Their foreign-citizen father went back home after divorce with adoptive-mother-that-granted-citizenship. Adoptive mother doesn't want anything to do with them anymore, not even provide basic documentation needed for State Dept to acknowledge their citizenship.