I am using WP-Members. I am able to add Social Login to the WP-Members user inscription page and create a new user with Social Login (or use the regular form). So far, so good.
However admin need to approve each user, options enabled by WP-Members with the proper email communication / flag on the website. Social Login completely bypass this.
Ideally, WP_Members and Social Login should integrate each other, there is probably a way to do it but I am not proficient enough to code it.
I am looking at the following workaround: users would have to register with the regular WP-Members form. Social Login would be presented only when the user will be fully registered/approved by admin.
EDIT: Found the solution - RTFM to myself
[oa_social_link] displays only when the user is logged in (and thus approved). Used this instead of [oa_social_login] on the user profile/inscription page.