I do like the idea of having to conversate purely in the public chat I feel like it could almost be a different game, instead of making direct alliances you'd have to play in the style of getting to know your housemates and evicting off that.
However the current fastings is great and a big part of that is due to private messages making the game what it is, a new game mode without the ability for private messages would be pretty cool however I can still imagine folk sending messages via the mail function and linking their chatzys which would defeat the purpose.