What they don't have, is the mandate to randomly stop people and demand "Papers!", like it's WWII Germany. This is what they are currently doing. In fact off duty Minneapolis police officers are reporting being stopped and told to identify by ICE/CBP officers. In every case it's race based. I expect there will soon be a Constitutional challenge, to their actions, based on this profiling.They are law enforcement officers, like US Marshalls and FBI agents. They can arrest people for federal crimes, obstruction, immigration violations, and any crime they witness. If it’s a state crime, they turn the offender over to state or local police.
They have challenges in sanctuary cities like Minneapolis, Seattle and Los Angeles because local law enforcement refuse cooperation. This makes it harder, slower, costlier, and more confrontational. When ICE doesn’t get support from local police they have to surge in a lot of resources to do the local police stuff - transport, holding, locating and identifying targets, as well as protecting their own officers.
They can't make claims for many of the crimes you listed they can enforce, when enforcement is predicated on illegal acts.
