Sorry you had to wait so long for an answer! It's not an easy question, and I don't know that I can offer much. Perhaps it helps to regard social media as just another kind of publicity channel and compare it with the much more familiar local newspaper. Would you entrust someone else to write press releases on behalf of the council? Probably not. Does it require special skills to write something for the local paper? Not really.
It's easy enough to get involved with Facebook, Twitter, etc. But I wouldn't think it mandatory if nobody feels comfortable with it. Or you could consider minimal use, such as simply a static Facebook page or tweeting the details of council meetings at the same time the agenda is released. The former could be set up by an outsider and approved by the council, then left unchanged.