Saruman was not fully faithful to the White Council
Robert T. Tally Jr. analyses the movie's portrayal of Saruman in relation to his more nuanced portrayal in the original Tolkien books.
In his brief appearance in The Hobbit trilogy, he comes off as a querulous, dismissive boss, who is already shown to be subverting the council he heads. For this reason, the “Leave Sauron to me!” — uttered in response to Elrond’s insistence that they pursue the temporarily banished Sauron into Mordor — appears as a dilatory tactic that would benefit Sauron, thus revealing to the viewers (if not yet to Gandalf, Galadriel, and Elrond) that Saruman is already a servant of the Dark Lord.
In The Silmarillion, Tolkien furnishes Saruman's motives at this stage.
Then the White Council was summoned; and Mithrandir urged them to swift deeds, but Curunír [Saruman] spoke against him, and counselled them to wait yet and to watch. ‘For I believe not,’ said he, ‘that the One will ever be found again in Middle-earth. Into Anduin it fell, and long ago, I deem, it was rolled to the Sea. There it shall lie until the end, when all this world is broken and the deeps are removed.' Therefore naught was done at that time, though Elrond's heart misgave him ... Thus the Wise were troubled, but none as yet perceived that Curunír had turned to dark thoughts and was already a traitor in heart: for he desired that he and no other should find the Great Ring, so that he might wield it himself and order all the world to his will. Too long he had studied the ways of Sauron in hope to defeat him, and now he envied him as a rival rather than hated his works. And he deemed that the Ring, which was Sauron's, would seek for its master as he became manifest once more; but if he were driven out again, then it would lie hid. Therefore he was willing to play with peril and let Sauron be for a time, hoping by his craft to forestall both his friends and the Enemy, when the Ring should appear.
The Silmarillion; OF THE RINGS OF POWER AND THE THIRD AGE
In summary, Saruman covets the One Ring and distracts the White Council from the pursuit and absolute defeat of Sauron in order to win a 'long game' by finding the One Ring first.
See also an answer furnished here on Sci-Fi Stack Exchange.