Germany actually does that! Quite a few government bodies are already active at https://social.bund.de/. Maybe there‘s hope that other countries will follow.

In some countries, corporations and government are basically the same entity. Free countries distinguish between them in a meaningful sense.

Absolutely. In fact I wrote exactly that last year.

Honestly, imo they wouldn't even need to get off Twitter and other platforms completely. Just make their own mastodon instance (or something similar that they control themselves) the primary source of truth and place of interaction. They could still link and reference it on other platforms to increase visibility, but make sure that all primary information is in a freely accessible place and not beholden to unreliable entities.