1. Entering a country without proper documentation is a crime. Therefore all "undocumented immigration" is by definition criminal.
2. Removing criminals is paramount to a safe society and a justice system that is respected.
3. "Documenting them and letting them live" undermines legal immigrants who likely worked very hard to integrate culturally, establish themselves, and do the proper LEGAL paperwork. These legal immigrants have stringent reporting requirements, need to be careful about even minor crimes (excessive speeding tickets even!) etc. How is your proposal remotely fair to them?
I don't understand why this is a controversial opinion at all. I have yet to meet a legal immigrant that isn't okay with booting anyone that isn't legal out. A country without border control is NOT a country.