No, that I get and agree with.
Getting into arguments with people on Twitter and taking his/her own pot shots like saying, maybe the NWS should put its briefings in coloring books is just stooping to Trump's own pettiness level. There's enough of that banter as is, and Trump seems to love it, that it gives him sort of validation to respond and demean whoever it is. You know he just ends up spinning it to say, "See, they're the bad guys America. I'm telling you, big, bad, guys."
I would expect a Twitter created as an alternative account of an official public-service page, to keep from being gagged as you say, to continue sharing the information they want to share, not repeatedly posting cute "sarcasm" gifs from Big Bang Theory in response to people like they're arguing on tumblr. The accounts can come across as almost parodies at times just b/c of the silliness they divulge in if Trump is brought up.