It doesn't follow logically at all. The origins of judicial immunity have distinct historical roots, just as Congressional speech & debate immunity has distinct roots.

There's an argument it might effect prosecutorial immunity, presuming someone was a stickler about consistency, but it's tough to say because the justifications are arguably based on a poor historical understanding of how prosecutors took up cases. (Historically prosecutors, even criminal prosecutors, were often private individuals paid by the local government to prosecute a case, or even prosecute a case entirely on their own volition. But today many jurists just assume the prosecutor has always and only been a distinct executive office, which if they're well read enough--but not too well read--might think sometimes had some quirks about who filled the office.)