"This is true; however keeping a website running is still very, very cheap compared to almost anything else the government does, including functions that are continuing as usual during the shutdown."

This is literally irrelevant when the non-essential services have to be shut down. If your techs get furloughed, shutting down the site is appropriate.

The twitter accounts are "shut down" in the sense that the employee who would have done the tweeting is now furloughed and can't. Putting out a tweet explaining the upcoming lapse makes a whole lot of sense to me.