What's the law that has teeth that can get them deported without adjudication? Is it for the accused to show they're here illegally or is it the judicial system? Does not having documents prove they're here illegally? Round em up all you want and they still get their day to be heard. While they're waiting for their day, show me what law allows for their detention without bond? "Build a Fence", on who's money and approved by who, President or Congress? Sure the Executive can do more at enforcement but it's the judiciary that determines if they get deported. Immigration Court is a minimum 4 months in the rears. Being an Illegal Immigrant is no worse than getting a speeding ticket in the eyes of the law because they're both just misdemeanors. Congress can make it a felony and that would help a lot by putting teeth into the law.
You are making several big assumptions:
1) no one (or some very small quantity) is admitting they snuck across the border,
2) there is no way that anyone can prove they lived in this country (via receipts, remembrance of addresses, places of birth, etc.) that could be used before adjudication.
I would think there's adjudication if there's a dispute.
But if Jose can't remember where he was born, or if there is no record online of his birth, or if there is no such address, and if he is using an invalid social security number (of a dead person say), then it's time for him to go back across the border.
And then, what about the illegal aliens that are visa overstayers -- why are they in this country? Your answer does not address this issue.
You must want them to stay here. Why do THEY get to set immigration policy. WHY CAN'T AMERICANS DETERMINE IT'S IMMIGRATION POLICY.