Of course moderators should have the right to put rules in the sidebar, and even remove posts that break the rules.
The notion, your notion, that mods' power is limited to enforcing publicly listed rules is plainly wrong. Arguments based on flawed reasoning aren't arguments at all. From reddit.com/wiki/faq:
If you have an issue with a moderator or the way a subreddit is being run, please first try contacting that moderator to see if it's just a simple misunderstanding. You may contact all of the moderators in a subreddit by messaging /r/[name of subreddit] to appeal a decision. Please keep in mind, however, that moderators are free to run their subreddits however they so choose so long as it is not breaking reddit's rules. So if it's simply an ideological issue you have or a personal vendetta against a moderator, consider making a new subreddit and shaping it the way you'd like rather than performing a sit-in and/or witch hunt.
Tell me, how do you imagine that the censorship the /r/games and /r/gaming moderators did was to enforce rules? You have evidence that they were deliberately colluding to delete content that they personally didn't like. That's not enforcing rules, that's being a poweruser.
Again, mods are not limited to enforcing established rules.
The solution to "bad moderation" is prescribed in the same FAQ answer: Instead of bitching and moaning, getting pissed off, staging a sit-in, or participating in a witch hunt, start your own subreddit.
The design of moderation on reddit is not a democracy, nor even a federal republic. It's plainly authoritarian. Moderators can do whatever they like, whenever they like, to any submission or comment in their subreddit, using any (or no) justification [so long as they comply with the site-wide rules]. A mod is well within their rights to delete a rule-following submission about a topic they personally don't want in their subreddit.
And enough with the cries of "censorship". That's not what is happening when a subreddit or mod prohibits a topic. Governments and states censor. Private entities do not.
TL;DR: Read and reread FAQ: Why does reddit need moderation? Can't you just let the voters decide? and FAQ: What if the moderators are bad? until it sinks in.