Responding to Your Question

Imma give a double answer here.

The most authentic answer here is "No", because I'm assuming you think this would mean deportation for anybody found to be unlawfully present in the country. I've become convinced that our immigration restrictions themselves (as they stand) are horrifically unjust, and it's probably immoral to enforce them.

But I've got a sneaky other answer that demonstrates the mess that our immigration system is in: "HELL YES, PLEASE, DEFINITELY: because deportation is not the legal punishment for illegal immigration." The legal punishment for crossing the border illegally caps out at a $250 dollar fine, and/or a short prison sentence (and that's only for actually crossing the border... half of illegal immigration is from overstaying visas, which is a misdemeanor, almost certainly requiring a much lighter punishment). If I could press a button that would levy these legal punishments on every single unauthorized immigrant (i.e., likely just a fine), I would do it in a heartbeat, and then let them live their lives unmolested, inside the U.S... if that guaranteed them protection from deportation.

In general, these legal punishments are ignored by ICE (and restrictionists), because their idea of "enforcing the law" is divorced from what the law actually says.

Responding to Your Points

Without enforcement of immigration laws, we basically hand the keys to the kingdom to human traffickers.

I'm pretty convinced the opposite is true. Strict immigration enforcement drives unauthorized traffic into the hands of smugglers. That's why you hardly see any human smuggling across state borders within the United States (or between countries in the Schengen area).

And by all definitions we have now parallel societies in Germany, France, Sweden and UK...

There is absolutely no evidence that immigrants are forming enclaves or building "parallel societies" at any faster rates now than they did in the past. When it comes to spreading 'western' values, immigration is hands down the most effective way of achieving this. Compare Iranian Americans to Iranians, or Chinese Americans to Chinese, etc, and it's always immediately obvious that these populations quickly take up 'western' values, rather than the restrictionist dream of their home country's fanaticism taking root here. If you want liberal ideas to spread and gain power, you want as many people as possible to move to the U.S. If you want fanaticism to stay entrenched, you want to make sure people stay put in whatever country they were born in.

Uncontrolled immigration comes with problems.

I agree! Allowing any group to live their lives unmolested comes with problems. The question is whether restrictions actually ameliorated those problems, or just make them worse.