Most European countries actually do in a limited fashion. Countries that have signed the Convention on the Reduction of Statelessness grant automatic citizenship at birth to people that would otherwise be born stateless.

More countries should adopt birthright citizenship. It has a lot of utility to it. It prevents the formation of a multigenerational undocumented underclass and greatly assists in the assimilation of immigrants into the broader culture. It's simply a fact of life that some immigrants will enter a country illegally. And while it is bad enough that they may live the rest of their lives in hiding, it's even worse when people are born into that condition. You can end up with generation after generation, people with little to no ties to their "homeland," living as a permanent underclass because they lack citizenship.

It's also a protection against some forms of tyranny and oppression. A favorite tool of tyrants is to strip citizenship from their victims. They'll sometimes go back generations and declare decades-old immigration cases as fraudulent or invalid. Look at the Rohingya genocide, where the Myanmar government declared an entire minority group to be illegal immigrants. Having a hard rule that says, "if you were born here, you have citizenship," prevents these tactics from being used on anyone except actual immigrants. Tyrants can still target immigrants, but their children are protected.

I looked into it when people were talking about Ted Cruz being born in Canada. His mother is a US citizen, so he’s actually a birthright citizen.