About the anchor babies argument:
For the people who are here and undocumented already, the anchor babies solution is overblown.

1) You have to wait until the child is 21 years old which is a long time to justify as a backup plan.

2) During the time the child is 21, you can still be deported AND the US govt says you can leave your baby here while YOU go back to your country since the child is a U.S citizen.

Unless you pray for your child to have a serious disease such that you can prove you have to be in the U.S with that child in order to make sure he/she gets proper treatment (and cancel your deportation removal), anchor babies are two decade long process and will not protect you from deportation.

For the people who have babies and then return to their countries thereby creating a backup plan, I honestly think there isn't enough data to prove how common this is as an option. Most times, Republicans or detractors refer to the people who are here now in the U.S who have children here as those with the "anchor babies."

If you end up adjusting your status legally through marriage and/or your child, you still did it legally (and we know there are some of us who fake the marriage for a green card) so politicians can't rail against something that is allowable by law.