> Pardon my skepticism, but what difference would that make to rename or reorganize DHS into a different shape?

ICE, being under DHS, is part of the US security apparatus. It has a threat-orientation. INS did have an enforcement component, but it was substantially an administrative agency. Immigration enforcement agents should primarily be process servers, notifying people whose papers aren't in order either what they need to do to fix them, or when their court date is.

Okay. Out of curiosity, in this arrangement, what should happen when these upstanding individuals, after overstaying their visa by a few years, simply don't show up to court or bring themselves into compliance, because they never intended to? Let's imagine for fun that they live in San Francisco, where the police are bound by local law to hide undocumented immigrants from the Federal government at all costs.

When someone doesn't show up for a required court appearance, the court issues a bench warrant, and they may be arrested, among other consequences: https://legalclarity.org/what-is-a-bench-warrant-and-what-ar...