“Rooms” in Matrix are more a space in which you persist arbitrary data rather than a literal chatroom. They can absolutely be world visible and used to create per-user timeline or per-tag timelines as you need for microblogging.
There are lots of different ways of skinning it. One we've thought of in the past is to have a room per sending user, and have the members of the room be that user's followers. The rooms are of course publicly visible to allow everyone else to follow along too. You the need some mechanism to index the messages based on hashtag etc, but that could just be a search engine - or possibly copying/referencing the messages from per-hashtag etc rooms.
