Just go to settings in the admin side and disable.
I agree the Groups page needs more redo. I do not care why. I would in addition not like to see the formal name of the group (the one with the underscore). I agree it should be a functional way to see the groups you are in and can join. It is also useful to show the ones you are not able to be in. For example, we have two groups of users that are special, Experts & Ambassadors. While people cannot join them, it is helpful to know who they are. Also good for staff listing.
Not for me. There is the ability to block from users who are not logged in. Also in the settings.