Why this app does not do private listening
Private listening means streaming the television's audio to your phone in sync with the picture, which is a fundamentally different job from sending button presses. The control channel this app uses carries commands, not media, and building an audio path would mean building a different product.
It would also be a bad trade for you. Roku already implements this properly in its own app, with the synchronisation handled by the platform that owns the audio. A third-party approximation would be worse and would exist mainly to avoid admitting the feature belongs elsewhere. So we say plainly: for headphones, use Roku's app.