Schala WAS the Arbiter. It's also possible Riddle was the Arbiter in the Radical Dreamers' timeline. However, once Schala stopped was sent to the DBT she stopped being the Arbiter. She was able to talk through the Flame due to her being a part of Lavos. She used to be the Arbiter, due to her exclusive power over the Mammon Machine. Once she disappeared, the Flame needed a new Arbiter. Needless to say, Schala just wanted to save her potential savior, not turn him to the new Arbiter. However, by the time of CC, Schala WASN'T the Arbiter.