ICE are the goons operating everywhere (i. e. not tied to the border) rounding up "suspected illegal immigrants".
Because it isn't exactly hard to find illegal immigrants, and their charter allows them to control people without objective cause, ICE gets to arbitrarily decide whom to harass.
It's the real-life version of the perennial fear of civil libertarians that too many criminal laws will just lead to any one of us being arrested whenever it happens to be convenient for whoever is currently in power.
It isn't going to end illegal immigration, nor curtail it to any significant degree. It just serves to keep a large segment of the people who see every day in a constant state of fear, unable to (for example) seek protection from crime or exploitation for fear of being deported.
As for the rest of your comment, this is the far-left extremist position that I cited in my first post. There's no good faith discussion to be had here.
No, it's the enforcement arm for immigration and customs, the former of which is people who are living in the US despite not being US citizens. Enforcement of people (and goods) crossing the border is the Border Patrol and it's parent organization Customs and Border Protection.
There's not much controversy over the people who cross illegally but stay temporarily and close to the border. Those are just traffickers and cartels.