Do the math on how many people are necessary to run a web site with on-call rotation, minimum moderation, and someone to run the business. The number of $2/month subscription required to make that work is prohbitively high.

Is this really so? Let's try doing the math: you're describing a distributed team of maybe 10 people, likely less. Let's assume you need $600K/year to run this business (is this the right number? not sure, feel free to correct me). At $2/month, that requires 25000 paying users.

Difficult, but not impossible. At $5/month (the $3 difference wouldn't trigger any price sensitivity, talking from real experience) that's 10000 users. If your service actually provides value, you can crank it even higher. Again, difficult, but totally within the realm of possible.

    if the unpaid social network competitors can't get any traction, what makes you think adding a $2/month signup hurdle would improve the situation?

Because the "traction" needed to make free and paid social network work is vastly different. You need insane scale (millions or tens of millions of users) to make free social network viable, hell, my stomach hurts from only thinking about it. A paid social network business, run with certain austerity, can be profitable with one thousand paid users.