Whether that happens because the organization reforms or because the organization ceases to exists is a secondary concern.
Government programs NEVER just end, and certainly not do to simple supply disruptions. So they won't STOP doing bad things, they'll just continue to do bad things with even less resources, making the net effects even worse. Eventually you might end up with ICE detention centers that would make a FOB in Iraq look like a palace. All you need is triple-strand concertina wire, some 55-gal drums cut in half (for burning human waste), and a bunch of plywood.
Law of unintended consequences....
In addition, "Government programs NEVER just end" is also obviously incorrect. Our government once had a program of segregating black people from large swaths of society. This program, thankfully, no longer exists.
If you had said "ICE would be forced to reconsider its inhumane methods which have lead to its terrible reputation" - a lot more people would agree with you. There have been a number of polls done over the past year or two which show that only about 25% support abolishing ICE, the majority oppose. I think almost everyone would agree that they need to do their job with a lot more empathy and humanity, so protests are good, but the goal should be to get them to clean up their act, it would be a mistake (and not supported by the people) to try to get rid of the immigrations and customs enforcement function entirely.