I guess you can say that their action was "semi-public". It wasn't trying to be completely private, and neither completely public, but somewhere in-between.
I am not sure you can, unless you play the politician bit and lie about "wanting to spend more time with family" or "health reasons".
If they resigned without any statement, people would have noticed as well, and speculation and gossiping would be still rampant, if not more rampant, because it's not every day that the entire Rust moderation team resigns.
They are between a rock and a hard place, if their objective really was to avoid public drama. Give too little detail and you're entirely at the mercy of speculation and maybe whatever the opposing party puts out (without other people even knowing who that opposing party is). Give too much information and you may irrevocably hurt people (like victims, or even the accused when a mob comes for them) or the community as a whole.
What would you have done in that case?
They did: What went wrong was that the mod team doesn't have the power to mod the core team.