What if you don't try to solve that but a different problem first? Make the app good enough for existing private teams to be usable!

E.g. we play weekly, organize the event through Facebook for years but still a pain sometimes. Somebody don't RSVP but still show up, so we are odd. Also Facebook events don't make it possible to limit the number of guests. If you make it really easy to use and useful enough such a level that existing private groups will use it (and they stick because it makes their life easier), new people will come.