The administration isn't arguing they should be able to search cell phones whenever. The case is about a legitimate search incident to an arrest. The key sentence from the article: "Last month the Obama administration stepped in and petitioned the Supreme Court to hear the case and rule that the cellphone search is, in fact, lawful, because, 'a cellphone is no different than any other object a suspect might be carrying' and thus fair game for searches."
Police are allowed to search you when they arrest you. That includes looking through say a backpack you might be carrying. The Obama administration is simply arguing that the contents of a cell phone should not be treated differently than the contents of a backpack. This is a sensible, intuitive argument to make.
The key consideration for such searches is securing weapons and preventing the suspect from destroying evidence. The weapon concern obviously doesn't exist with cell phones, but the destruction of evidence concern certainly does.
Obama's argument is that creating the additional wrinkle of disallowing searches of particular kinds of things will create additional litigation over searches incident to arrest. Which is a legitimate concern: the courts are filled with all sorts of nonsense by criminal defendants trying to get off on the technicalities of searches.