The expected value for immigrants is rapidly shifting into it being more favorable to be an illegal, because ICE/CBP is mostly going after low hanging fruit of easy to catch people that they know about with homes and salary jobs / university and a visa. People that are off paper and 100 miles past the border are as good as gone. So basically what we get is the exact opposite of what we want.
Your own premise destroys your argument. If they're grabbing 'whoever' it's at least as easy to grab immigrants with a paper record as those that don't.