Yolo it would be technically very challenging to disable the entire profile page, because it's technically just a frontend for searching one user via the API, disabling which would likely break many extensions.
However, each extension can easily provide a permission for its content on the user profile, so you should for example be able to hide the bio from guests. If an extension adds content but doesn't have a view permission yet, it should be relatively easy to implement.
It should also not be too complicated to disable searching discussions and posts by users if you wanted to hide those tabs and make it impossible to access via the API. But someone could still brute-force the API to get all discussions or posts and iteratively filter them by user if the author of posts and discussions isn't hidden as well.
