I think there's a false dichotomy here between solving a problem and building an ad-supported service. Facebook definitely solves a number of problems. How to keep in touch with people, communicate, share photos, etc. That's why people use it. Admittedly it doesn't do anything that couldn't be done (or wasn't done) another way before it existed, but it reduces the friction and wraps them up into a great package.
If free social networks were banned, a lot of people would still pay $5 or $10/mo to use Facebook. But since a law like that seems unlikely in most countries, people will always gravitate to a free solution or some other method of monetization more creative than direct payment has to win.
I don't know about $5 to $10/month for Facebook. I would like to see a HN vote/survey on how much people would pay per month for the most used web properties if paying was the only way to see them. For me, the maximum monthly costs would be:
NYT. $15
Gmail $3
LinkedIn $2
Facebook $2
Not sure though about Facebook, HN, Reddit, etc. since they would be so much less valuable without large user bases.
If free social networks were banned, a lot of people would still pay $5 or $10/mo to use Facebook. But since a law like that seems unlikely in most countries, people will always gravitate to a free solution or some other method of monetization more creative than direct payment has to win.