It'd be great if you could just hide it to anyone who's not involved in hosting you or being hosted by you... I really don't like just anyone being able to see where I'm going, where I've been and with whom for security reasons. Make the feature available only to block the rest of the people from seeing your trip history if they are not concerned in an airbnb deal with you and they haven't asked for permission.
They would be able to see your reviews if they happend to be interested in renting a place where you've been before, but they can't find that place through your profile because they can't access your trip history unless you allow it. I think this would solve many of this conversation's participants' issues with the platform, including mine, without affecting Airbnb's functionalities.