In most tabletop and skirmish games I've played faction defines your team. But what if it instead defined the individuals that make up your team? Could this even work in a skirmish game? Maybe it's too much Fallout talking, but this seems like it could be a good twist for a post-apoc game.
Last Days: Zombie Apocalypse might be similar to what you're referring to.
There's three "factions/alignments"; Selfless, Selfish, and Trained, each with their own starting leader character (The Everyday Hero, The Merciless Thug, and The Professional). There are model types that align with these - a Gang Member is Selfish, a Soldier is Trained, a Paramedic is Selfless etc. There are also Neutral models, like a Kid, Dog, or "Survivor".
Traditional group creation is that each group has to have a minimum percentage (half?) of models whose type aligns with the leader, and no more than a quarter can have a different alignment to the leader - you can take as many Neutral models as you want to fill out your group otherwise.
There is also a supplement called Chaos Theory, that allows for random group composition that can break the normal group restrictions at gang creation (a Merciless Thug surrounded by Selfless people, for example). It can also be used mid-campaign, but it means you don't recruit who you want, you get the opportunity to recruit whoever randomly turns up.
As a post-game action, the leader can persuade a model with a different alignment to change to the leader's alignment, so you can over time get more models of a different alignment in your group (make a Gang Member believe in working for the greater good of humanity, persuading a paramedic that they should take advantage of the apocalypse and screw everyone else, convince a Survivor they've got what it takes to be in the military, etc).
From a "faction" perspective, you can have anyone or anything using those rules:
A biker gang that survived the Apocalypse is generally styled as Selfish in LD:ZA, but it's mini-agnostic, so your Everyday Hero could be a biker with a heart of gold that is slowly finding and persuading other bikers to work for the betterment of humanity.
A Trained group would usually be military or police, but your Merciless Thug could be a corrupt cop that is showing his fellow officers that in this new world, might is right, and they have all the training to take it.
Your Trained group might also be a bunch of doomsday preppers that follow a military command structure, working out of their camping site in the woods.
Your more "traditional" factions, like Goliath, Orlock etc. have a place in games in different ways, like the skill access as mentioned before.
Stargrave and Frostgrave haven't got "factions", but you do have your leader/wizard and second-in-command/apprentice that can be the "flavour" of the group or influence the group composition.
In terms of factions as a playing mechanic, it still has legs, but only if it doesn't hamstring the game; letting players branch out from those "factions" should be possible to keep the game interesting, but by no means should they be thrown out as old hat - at a basic level they provide a solid basis for people to start playing and experience different play styles.