Rizo playing his fake idol instead of his real one has the same effect as Rachel playing her Shot in the Dark on 47. It allowed him to gauge reactions. He could tell from her reaction that Yellow Sophie voted with his alliance, so he didn’t need to play his real idol.
You may find him annoying but he’s playing the game at a much higher level than most of the rest of this cast.
Rizo might be playing his game at a high level, but I think he's blind to some of the threats around him. He and Stephen are the only males left in the game. I can see Savannah proposing that all the women vote together (after Sage and maybe Kristina are gone) and have an all female final 3. Savannah comes across as a Parvati wannabe that uses men in the game to further her agenda and then casts them aside. She bats her eyelashes at Rizo and he does her bidding.
I can't tell if Savannah is aiming for a final 3 of Soph, Sophie and herself or Soph, Kristina and herself because Sophie is too good at challenges.