The ice class requirements are being driven by politics and safety theatre. The conditions shown in the video are milder than what the fishing boats with no ice class deal with around the Pribilofs every winter.
If they set the drilling season from around July 1st when the ice is usually gone until ice usually arrives around October 15th, no ice class would be needed. It seems ridiculous to build a fleet of ice class vessels to extend the drilling season by two or three weeks before they find any oil. If they do find a lot of oil, they are going to need rigs and support vessels capable of operating a longer season.