Republic is best. Anarchy is worst. That is the essence of it. Avoiding unnecessary anarchy tends to be the major consideration. So for simplicity just stick to to despotism -> anarchy -> republic.
While there are times when monarchy is slightly better than republic those times tend to be short and the advantages of republic in the remaining time are significant.
I'd say it would be situationally dependent. Most of your play, at least that we hear about, is the min/maxing form of play where you're permanently attempting to improve your game regardless if you actually need to or not. In your case, the maximising is your need, but if a player is just playing for random abstract fun then that isn't a need.
In terms of abstract fun games, Republic does have it's own very severe discomforts. Managing War Weariness is extremely tedious from a relaxed gaming point of view, and is more so on a Pangea than on an Island. The very nature of Republic tends to force you into not engaging in warfare in any serious way until you reach the Industrial age, so if you fancy just spamming Horses from the Ancient Age onwards, again particularly on a Pangea or large Continent, then both Despotism and Monarchy are heaps more relaxing.
When talking about Archipelago, or a situation where you've got a gazillion cities all with horrendous Corruption and you're playing the game in a more simulationist way by imagining you're a real life leader wanting to make a 'nice' empire, then converting to Communism makes for a much more pleasant gaming experience as you can build, build build to your heart's content with the added bonus of no War Weariness.
Then when talking about Tiny Pangea maps, then you can clear out everyone up to Emperor level difficulty just by spamming Archers in a Despotism.
The ones I never use or disliked when I did were: Feudalism, Democracy and Fascism. Feudalism only really has any use if you strangle all your cities to size 6, and then you still get War Weariness from your larger army. I'm sure there must be an ideal scenario for it, but I never found it. Like r16 says, I never noticed any benefits at all from Democracy even though the game implies there's supposed to be a big deal involved, really bizarre design decision there, it might be, like, 5% better than Republic but at the cost of the most hideous War Weariness imaginable? Fascism just seems ugly, and if you're playing for a war-spam game then you could have done that with Monarchy and if you're still at it by the time fascism comes around then you're probably not very good at warring anyway.
