You've piqued my interest. I'll definitely spend a bit of time considering it.
My favorite broad structure is to have a stable of preset heroes that represent each faction - even, perhaps, creating multiples for each faction to allow for diverse styles and to allow for more cameos!
The objective would be to achieve specific victory conditions that align with your faction, each granting a victory point. Some amount of accumulated victory points would end the game crowning that player the victor.
What could make this interesting is that these tokens - the victory points themselves - could be on conditions that are a) shared among 1-3 factions, allowing for a minor amount of cooperation and/or b) be competitive conditions, where to have one nullifies the ability to gain another in an opposed faction. These could be dynamic tokens, such as struggling over control of the Tower of Pain.
The key to making it work is to make a good list of cooperative and competitive goals that would merit a token and align them with a faction. This sets up our objective, but nothing more.
What you need now is a Resource system that allows for things like trading and faction-based perks, and a random Encounter system that also interacts with resources and factions. At first blush these should be card decks, but they could also be present on tiles.
Probably MOST important, however, is designing it such that player input is relevant, and it is not a procedurally solved game of simple RNG. Trading is one way to do this, but the avenue of basic resource generation should require some special effort of the player in order to keep it "gamey", not something to simply watch unfold.
I'm just spitballing thoughts, not offering solutions. And anyway, I already took the wheel when it wasn't desired.