I only support this then if twitch chat would be visible in-game.By this I think you mean that you could monitor your twitch chat in game with something like a #stream channel? If so, then yes this would be an amazing idea, it'd make it a lot easier to monitor and reply to chat with this. I also know for definite that the twitch chat works with IRC clients and so I can only assume that it'd be possible to integrate into the game.
EDIT: Also Thor, surely integrating this chat would be easier than the game setting up a stream for the user? Software like OBS are not hard to use, and there are even countless guides out there to help one set up their stream on these software, so it'd just be a waste of time and resources to implement it within Osu! itself.