Can't speak for everyone, but mine's a bit more nuanced.
There's little cost to it, but little cost to changing it either, as long as other parts of law are changed to cover the gap made by simply doing away with birthright citizenship.
The problem is, we'd create perpetual non-citizens (as others have mentioned). Places that don't have birthright citizenship almost always have some modified method (such as can become a citizen at 18 if you've lived here x years as a minor, or similar) that prevents this permanent group of uncitizens. I have no issue getting rid of birthright if we implemented something similar.