And if she's got nothing else to do at the moment and feels like playing solitaire or something, who gives a shit; having entertainment during downtime puts workers in a better mood, and hence it increases productivity

Social media could be problematic I suppose; some people would feel too entitled to just talk on facebook 90% of the time and would act like you're inconveniencing them when you ask them to actually do some work.
Most people, if they actually enjoy their job and are given the freedom to be creative, aren't ever going to abuse their freedom though (given that they actually have something to do and are never being pressured to always be doing something).

And anyways, with smart phones nowadays, you're never going to totally stop people from going on facebook and whatever when they shouldn't.

Besides that, there are legitimate reasons someone might need to go on facebook during work: they might need to ask their husband/wife to pick up their kid from school; they might need to look up a phone number that is only listed on facebook; etc..

Again, you should stop thinking 'my job is to keep things as difficult as possible'