I've managed to find a solution to my problem. I solved it with three plugins
The combination of these plugins allows you to who can log in, without telling them how to log in. More specifically, anyone will be able to log in with a social network, but they won't be able to see everything. With User Role Editor, you can make a new role based on Subscriber. I called it 'Unknown'. Set new users to get this role, instead of Subscriber.
Now, with the Members plugin, you can determine what roles get to see a page/post. I set it for everyone by default (when you don't check any checkboxes). but if I want a 'members only' page, I can check the necessary checkboxes, and 'Unkown' users won't be able to see this page (or get a message that they don't have access to this page).
Finally, the Social Login plugin allows people to login with Facebook, Twitter, etc. It requires a registration with OneAll, but I found that to be fairly user-friendly and quickly set up. Now, a new user can log in with Twitter for example. He/she will be added to the users of our WordPress site, but will have a role of Unknown. Now I can verify if this is a user known to our sports club and assign the role of 'Subscriber' if necessary.
Presto! Choose who can see certain content, not how they get to that content.