the tv has no control over splitscreen. it takes the video feed from the xbox and displays it only. the xbox (well the game running on the xbox really) is responsible for rendering splitscreen.
minecraft is a downloaded content correct? my guess would be either a new patch was implemented which has a split screen bug or the minecraft files became corrupted. in which case you may want to try reinstalling. if minecraft is not a locally installed game and is server based perhaps its a bug after an update on their end.