Do you mean something like a lifeinvader / IC Social network?
If so it's been tried and tested multiple times and in my experience, it just counts for tonnes more metagaming and a non-administrated battleground. If properly administrated and users treat it like a social network then it can work.
Management said they won't get involved in a social media but don't oppose one being made. So it's up to a member to make one, but it costs money to have a social network software and to host the website. Personally I would love to see a social media for Bay Area, a mix of Twitter/Facebook would be ideal. On roleplay servers, people add each other on IC social medias for the sole purpose of following their roleplay and not necessarily because they're friends, so I think Twitter follows, reposts + Facebook's private messaging, post likes/reactions and user personal info fields (employment, education, male, gender etc) and business pages would be a perfect combination.