Since Im a semi lay expert on Immigration law, Ill answer the question.
Since the INS has its own prosecutors and Judges and court system, these cases will not tie up any other Local, State or Federal court. So all the other crimes will still be handled as usual.
The vast majority of these cases will be handled administratively, as provided by Immigration law, so the alien can plead not guilty all he wants, and never see a lawyer or judge.
Under the Immigration Law, most of these illegal would be charged with a violation of the INA (Immigration and Nationality Act) that does not allow the alien a lawyer or hearing. Im the prosecutor, judge and jury, and I do this daily.
For those that are eligible for a hearing, they would pose a problem with detaining them and tying up the Immigration court system.
If so, we have a lot worse problem with non-enforcement of law than immigration law. I suspect that you mean that you have some civil power under administrative law to exclude from the country people who are foreign nationals or to send back to their land of origin those who have illegally entered. Even that is highly questionable inder the due process clause of the Fifth Amendment but it would be yet another erosion of the rule of law.
These questions are posed respectfully and not argumentatively.
The fact remains, in any event that resources get tied up and particularly if incarceration is possible. If those crossing the border determine to take whatever consequences are dished out, their entry is still essentially unstoppable. As you have said elsewhere, if someone calls INS criminal enforcement, they will get an answering machine and likely no action.
Also, please clarify your last sentence, if you will. Do you mean that the volume of cases even in the INS courts would tie up the system?
Finally, I think you suggested previously that the government would not encourage private groups to act to enforce laws that the government itself does not want enforced. If so, is that not a concession that, at least, the government lacks the political will to act to enforce the immigration laws strictly?