Someone who's in a communications or social media position likely does, and they should have a dedicated device purely for that function.
They don't. There was a time when government eschewed social media entirely even though it was available. They're not private businesses and I see no reason for them to be using social media in any official capacity. There are plenty of ways to communicate with your voter base that doesn't involve Twitter battles or ridiculous Facebook propaganda posts.
Also government usage of social media aids the flow of misinformation, by blending info from trusted providers (the govt, in theory, lol) with info from untrusted providers.
There's a reason the .gov domain exists.
That’s like saying they don’t need ears. It’s a place to see what people are talking about. It’s a place to see how people are talking. It’s a place to gauge responses to certain events. It’s increasingly a place to see news happen before it is shared via any other medium. You go where the people are, you don’t tell them where to go and then get upset when they’re not there.