Without the rape allegations, what crime would have led to the UK arresting Assange, enabling him to be extradited? Even with the allegations, the process of extraditing him to Sweden took ages.
He can be arrested directly as a result of an extradition request. (If this weren't possible, it would rarely be possible to extradite anyone.) I think you're perhaps unaware that an extradition request is required to present evidence that the person in question has done something that is a crime under both US and UK law. It is not merely a request to move someone from one country to another.
