> 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.