Why does Gandalf start speaking in Black Tongue at the Council of Elrond?

First of all, when Gandalf was speaking the black tongue he actually quoted the ring inscription, which is itself written in black tongue. So for one point he "stayed true to the source material":

                          Ash nazg durbatulûk,                            One Ring to rule them all,
                           ash nazg gimbatul,                                 One Ring to find them,
                          ash nazg thrakatulûk,                           One Ring to bring them all
                      agh burzum-ishi krimpatul.                    and in the darkness bind them

But at this point it had multiple purposes to really say it in black tongue. When he did so everybody was shocked to hear this spoken there, which is in fact what Gandalf wanted to achieve. Because at this point Boromir was standing up and nearing the ring (seeming as if dragged to it by the ring itself), about to make a statement for using it to fight Sauron. So recitating the ring inscription in black tongue was to remind him and everybody else that "the ring is alltogether evil" and can never be controlled. And using the black speech gave his words the necessary power and effect to convey this message unmistakably.

And last but not least using the black tongue openly instead of silently off the record also demonstrated that Gandalf is not closing his eyes from the evil that is already growing in middle earth and that everybody else seemed to ignore (and that might take overhand if kept ignored, which he was saying with the quote from your question). (similar to Harry Potter being for a long time the only one to call "you know who" by his actual name, to draw a connection to another famous movie franchise.)