Even Martial Arts has Twitter, and they only get 1 question in 3 days.
This is curious. What prevents SO and MSO from Tweeting?
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Your ideas actually aren't that misplaced. Don't get me wrong, Stack Overflow doesn't need its own Twitter account. In fact, I'd almost venture to say that none of the sites really need their own Twitter accounts. However, having a human Tweeting questions is a good thing that will benefit the community a lot more than a bot will.
Why? Because it's a bot. How many automated Twitter bots would you say you follow? How many are you excited to receive spam from? The fact is, bots just aren't that exciting. Maybe their owners are excited about them, but the actual general population probably is uninterested.
However, if you're a great programmer and have a lot of programmer friends who follow you on Twitter, then when you find and share interesting questions, chances are that will set off your Twitter friends' radar, and chances are they may decide to see what you find interesting. You're a person. People are interesting. Bots are not.
So, in short, we have millions of users who could possibly be Tweeting questions, and this scales much better than a bot. Hope this helps!