tl;dr version: May elected moderators on a specific Stack Exchange site ban users for doing things that the site's community disfavors but that are not against the Code of Conduct?
The Code of Conduct has been (arguably) hashed to death over the past few months, but all that discussion got me to wondering if it is truly the be-all and end-all of conduct rules.
It seems pretty clear that the Code of Conduct (CoC) is mandatory on all Stack Exchange sites and cannot be dispensed with at the will of the community, but what about the opposite scenario? If a community wishes to add conduct rules not included in the CoC, and has the support of the elected community moderators, may the community moderators use their tools to enforce said rules of conduct?
To be clear, I'm well aware that much of the Code of Conduct is contextual and thus many CoC adjudications will depend on local conditions. I'm not talking about that, but about a community coming up with brand-new substantive rules. For example, the current CoC requires everyone to use stated pronouns if known, but does not require ordinary users to affirmatively investigate or inquire into a user's pronouns before making an educated guess. The rule only applies once a user has been made aware of specific pronouns and knowingly and intentionally refuses to use them. If a site wanted to make it mandatory for users to ask for pronouns in advance, would the site's diamond community moderators have the authority to suspend the accounts of users who refuse to do that but still obey the CoC, or would this be considered an abusive misuse of diamond moderator powers, possibly subjecting them to removal?
I'm also not talking about close reasons - it is well-known that the community has some discretion on what closure rules will exist on the site.
As a comment alluded to, it seems obvious at first that the answer must be "yes", but it did occur to me that this could be used in an abusive manner. A site that wanted to keep certain people away could impose arbitrary or discriminatory rules and then "set people up" for suspension (e.g. "It is the will of the community that no user may participate who has not uploaded a current, untouched picture of themselves as their profile picture. Users violating this policy will be suspended for the maximum time that a community moderator is permitted to impose. Asking questions about this policy will result in a six month suspension."). Limiting suspensions to CoC violations could possibly protect unpopular users and users who have differing viewpoints or practices from those who hold social prominence on a site.
Another way of phrasing the same question is whether "I didn't violate the CoC" is a defense against a suspension. For example, if someone contacted the Community Team and said "A community moderator at Stuff.SE suspended my account because they said I violated their rule against having political speech in my profile, but having political speech in my profile is not a violation of the overall Stack Exchange Code of Conduct as long as it is done in a civil manner. Please reinstate my account and warn the moderator(s) not to do this again.", would they actually do anything?