A friend of mine, C, had an interesting experience:
Her parents, refugees from the Vietnam war, resided in Switzerland when C was born. C has a Swiss birth certificate, but is not a Swiss citizen as Switzerland does not grant birthright citizenship. C's parents immigrated before C satisfied the Swiss residency requirement; C became naturalized as a US citizen 25+ years later. Her experience is interesting to me because, until a few years ago, she was not a citizen of any country.
Many USians do not believe that children of unnaturalized immigrants should have birthright citizenship. I have many coworkers from India, Nepal, China, and elsewhere who reside in the US legally on a work visa; many of Indian coworkers wait for 12+ years before they become naturalized. In the mean time, they may have children who, if not for the benefit of birthright citizenship, would not belong to any country. Suppose that the US did not have birthright citizenship, and the parents of a child lost their visa status for one reason or another (e.g. this can happen of someone on H-1B status is laid off and remain unemployed for at least 3 months); what would happen to their children? They can't be deported back to India, they also can't get a job in US, nor attend school, nor even apply for a passport. They would literally be trapped in the US, and would have to navigate the US's legal structure in the same way that illegal immigrants do (despite being neither illegal nor an immigrant).
Birthright citizenship protects people who might fall into the class of stateless people. I'm not sure what the alternatives are that would not seriously disadvantage children on of unnaturalized US residents.