BuddyPress.org

A couple ways to do this I think. One is to just redirect people that aren’t logged in to your login page. That way if they hit the home page, they’ll just get sent to the login page if they’re not logged in. There’s a few recent threads with ways to do it and some plugins I think also. Then, you’ll want to exclude the login page from the Nav too in your theme.

It seems like you could also do this with a child theme where you just check is_user_logged_in() and based on that display the activity or login. I haven’t looked into this much, but it seems reasonable. There’s probably an activity template you could include using locate_template()