This is just speculation, but what if Rosenstein has seen evidence of extensive collusion or other criminal activity by the president or his associates that he believes may lead to a large number of indictments that will basically topple the Trump administration?  What would you?  Go public immediately and compromise the investigations?  Or play along and in the process get rid of the person in charge of the investigations (Comey) that, to many on the right and the left, appeared to be compromised because of his actions concerning the Clinton email server?  Then appoint a person of impeccable credentials and reputation {Meuller) to led the investigation which may result in the resignation or impeachment of the president.  If Rosenstein is a man of integrity this scenario, or something like it, might explain his actions.