So what you really want here is a change in laws such that we automatically offer a blanket amnesty to illegal immigrants.

Exactly not.

This is the rhetorical shift that the Administration needs, and exactly the problem with denying due process to people. Because we reach different conclusions when talking about an entire group of people than we do when considering a specific individual case. Again, to borrow from Les Miserable, people can agree both that the law is important and should be followed and that what Javert was doing was not appropriate and wasn't justice.

How you choose to deport people matters. Allowing them to exercise the rights that they are entitled to, to go in front of a judge and argue that there are humanitarian or other reasons specific to them why they should be given some consideration or accommodation. In the specific case we're talking about, it is absolutely unconscionable that at no point in the process was any consideration given to the specific fact that this woman had an infant. Even if she had been convicted of an actual crime in the U.S. criminal justice system, her lawyers would have been given the opportunity to go in front of a judge and argue for accommodations in her sentencing to take into account the needs of her child (certainly delaying the commencement of her sentence for a brief period to make arrangements).

Of course, it also matters on the substance as well. Most Americans don't want bad guys coming here illegally, and are happy to throw out the bad guys. But while they are comfortable throwing out people who are malicious or violent, they are more reluctant when it comes to cases of people who have proven after they arrived that they are actually upright and virtuous people. They want control over the border so that we don't inadvertently end up with malicious or violent people coming in, but once we know that someone is upright and virtuous after spending many years here, they reach a very different balance of the equities when they see what happens to them in the system. Which is why Trump is losing support in polls on immigration, even though he's been enormously successful in driving down border crossings - because people are uncomfortable once you start deporting the harmless abuela or the hardworking busboy at the local cafe.

That's why Stephen Miller will always pivot back to talking about millions of people - and why you shift away from talking about any specific case, and whether what's happened to this specific woman and her child is appropriate or fair. There's absolutely no reason why she shouldn't have been given some amount of time to sit down with her husband and figure out what was best for the child, whether to have to live under Cuban dictatorship or be separated forever from her mother. Forcing them to make that choice is already pretty damaging, but making them do it in the space of a few minutes when there was absolutely no reason for that is just unconscionable.