The Fourteenth Amendment granted citizenship to all persons born or naturalized in the United States. It was further codified by the 1965 Immigration Act. It'd take more than Trump to repeal birthright citizenship.
I think it is a bit more complicated than that, but ultimately, you're right.
First, I don't think it is certain that the 14th Amendment was originally intended to apply to the children of illegal immigrants, and the Supreme Court has never addressed that question squarely. People cite to
Wong Kim Ark in support of that, but the parents in that case were Chinese citizens who were here
legally, so any really broad language that you'd try to apply to the children of people here illegally would be non-binding dicta. So maybe there is an argument there, although I think that decision being much closer in time to the passage of the 14th is pretty good evidence as to how it was understood originally.
However, I think the Immigration Act of 1965 closes the door on fixing that with an Executive Order, because as we all know, an Executive Order can't override an actual law. The Immigration Act of 1965 does use the exact "subject to the jurisdiction thereof" language as the 14th Amendment, so you could try to argue that it is equally ambiguous and therefore doesn't necessarily bar an executive order.
The problem I see with that argument is that there was nearly 100 years of the 14th Amendment's "subject to the jurisdiction thereof" clause being interpreted to apply to the children of illegal immigrants, and Congress certainly would have known that when drafting, debating, and passing the 1965 Immigration Act. So, when Congress passed the 1965 Immigration Act, they were pretty clearly ratifying the broader language of Wong Kim Ark that applied birthright citizens to everyone except diplomats, etc..
In other words, even if the original understanding of the 14th Amendment didn't cover the children of illegal immigrants, the original understanding of the Immigration Act of 1965 did. And Trump can't override that law with an Executive Order.
I'm pretty sure that's exactly what the Supreme Court would say if it made it up to them, which I don't think it would. So if you really want to challenge in the courts birthright citizenship for the children of illegals, you'd need to get Congress to pass that law. Otherwise, the courts would never even reach the Constitutional issue, and would just cite to the 1965 Immigration Act.