This is primarily for tabletop, but it can also relate to computer games, but in my faint hope of starting my sessions back up, I'm theorizing a system where factions are given a score that the players can affect.

Say the party is working for one faction, and their success and failures directly impact the score of that faction. I don't want to give the impression that my players are doing the heavy lifting for that faction, so other NPCs are contributing to the score. And other factions are given their own score with their own NPC actions. My first thought was to do some 2d10 rolls, but I've not yet thought how that would affect the scores, so my other thought is a little game I came up with that plays itself with tokens and dice rolls, albeit modified to reflect the actions of the players and the probable actions of the overall world.

Basically, the players' successes would either add to their faction's score or subtract from enemy factions. Their failures would do the opposite, subtracting from their faction's or adding to enemy factions, depending on what they've done. Likewise for the NPC side of things. The scoring system isn't a reflection of every action taken, just more a way for the story to have stuff happen away from players. "While the party was busy doing x, y happened." Then motivations and consequences for factions can arise from that situation (a faction dissolves, creating a bandit problem for the world. Two weakened rival factions merge. Factions start preying on the weaker ones, etc.). The intent of all this is to weave in active worldbuilding in an open-ended campaign.

Anyone have any decent ideas to go about this?

Check out Blades in the Dark RPG (or its variants, like Band of Blades), the concept of "fronts" from the Dungeon World RPG, the Fate Adversary Toolkit [very Fate-centric if you're not familiar with it though], or adapt the campaign structure (timeline sheet) introduced for Fate of Cthulhu [still Fate-centric, but probably more adaptable].

Or, you could treat the villains as normal characters and adapt some of the campaign systems from Pathfinder Ultimate Campaign (the sourcebook that led to the Kingmaker adventure path): http://legacy.aonprd.com/ultimateCampaign/campaignSystems.html

I'm sure I could think of more, but these were the first ones to come to mind.

While this is useful, skimming through some of this it unfortunately does not answer my question since they seem to pertain to micro-level encounters when I'm trying to devise a way of semi-randomly determined status of strategic powers. For example, I had an idea for a quest, and I wasn't sure if the NPC in question would succeed or not in their task, so I rolled and they failed and that kickstarted the rest of the quest events. But here, I'm trying to determine on a macro level what things might happen, but in abstracted in such a way that it could just be "bad thing happened" and then I decide what that bad thing was, and then the players have to deal with that. What I'm trying to figure out is how to organize all that and get some perspective on things I may very well miss.

If you can explain in a nutshell what these systems can help with in that particular aspect, I'd be all ears, it just seems like I already have most of that covered anyway.

All of my suggestions, except perhaps Adversity Toolkit, are precisely what you're looking for. They're not encounter-level things but background-plotting things. Dungeon World's "fronts" might be most. PF1 UC's are very mechanical, but are also very "meta"/background and not encounter-level. (It goes beyond the specific page I linked: the downtime page also has buildings and organizations, there are also the kingdom-level rules.)

I actually did read through the fronts section of Dungeon World, and that is closer to what I was looking for, but again, I was merely skimming and may have a closer look later. Everything else was not obviously available after a quick search. I thank you regardless if it's actually what I'm looking for, it will be very useful for future sessions.