Is there a way to stop routing LLM output to the TTS node entirely (not just mute playback) so that TTS is never invoked when sound is off?

Have you tried hooking the tts_node? Something like this:

async def tts_node(self, text: AsyncIterable[str], model_settings: ModelSettings):  
    # Check if sound is enabled (you'll need to track this state)  
    if not self.sound_enabled:  
        # Return early without invoking TTS  
        return None  
      
    # Call default TTS node when sound is enabled  
    return Agent.default.tts_node(self, text, model_settings)