Regarding what government does, it doesn't matter if a thing is "best", or "legal", or "logical", the only thing that matters is "the way things are".
Libertarians, despite any logic they may bring up on the subject, will never pursuade government to legalize drugs. Because drugs being illegal are "the way things are".
Likewise, you and the millions of Americans who feel this way about illegal aliens, will never pursuade government to truly shut the border, and deport millions of people. You're wasting your breath.
I'd prefer they were gone too. But since that won't happen, the next best thing is to get them assimilated. You are wasting your time, and actually getting in the way of assimilation, by trying to get them sent home.
The status quo is the worst of all situations, where illegals keep coming across, and never assimilate. The "no amnesty" mantra folks are actually helping purpetuate the status quo. Because these people won't be sent home. Assimilation is the best option we've got.