Arctic and Antarctic icebreaking with nuclear powered icebreakers should be done by the US Navy (not the USCG). Building and operating a nuclear powered ship is beyond the current capability of the USCG and they know that.
The Navy has the nuclear expertise with both submarines and surface ships, and a good safety record. The Navy and its contractors also know how to build nuclear powered ships with high strength steel.
I suppose the mission is to project sovereignty over the mineral rich Alaskan continental shelf and to be able to conduct research and freedom of navigation missions anywhere in the Arctic at any time of year.
As shipping increases in the Northwest Passage there may be more ice breaking assistance to shipping, SAR, and environmental protection missions.
Alaska doesn’t have many icebreaking requirements. Valdez is a virtually ice free port. The big tides break the ice in Cook Inlet. Western Alaska and the North Slope only have shipping during the increasingly long ice free, or nearly so, season. The USCG does not do ice breaking to support commercial shipping in Alaska.
The Great Lakes, Mississippi River system, and the Northeast are the USCG’s primary ice breaking missions in support of commercial shipping.