Well, if they want to try that, if they came here illegally, we can ship 'em back illegally if Mexico doesn't cooperate.
There are thousands of deportable aliens in the US who have committed aggravated felonies and will never qualify for a green card. The US Supreme Court has already decided that such aliens must be released into US and given work permits. If the home country refuses to take an alien, the alien must be paroled as a de facto legal immigrant.
And those are an exception of thousands in a population of millions. Sorry, but we aren't gonna hold back the process because of exceptions. That is an obstruction method.
We can, but we may not. The US Supreme Court disagrees with your statement.
And US Supreme Court's decisions are the law of the land, not the exception.