Treat the organisation like a character. Give it a stress track and consequence slots (if it runs out of stress and can't absorb it with consequences then it is taken out and becomes the smouldering remains you mention in the question). Then give it stunts and/or skills of its own.
As it improves, it gains stunts / skills. As it gets weaker it takes stress and then consequences.
Organisation Stunts can enhance its own skills (if it has any) or enhance the players' skills when they use the resources of or act on behalf of the organisation.
If it has skills of its own, then players can perform Create An Advantage actions to generate aspects that can boost the organisation (this keeps the player characters relevant).
Since it works in exactly the same way as any other character, there aren't any new rules to learn. It just means applying the rules you know already in a slightly different way.
Purple Tiger Mercenary Group
- Warfare: Fair
- Intelligence: Fair
- Influence: Average
- Wealth: Average
- Asari PR Officer: +2 bonus when negotating with Asari clients
- Mild [___________________]
(NB: Intelligence here is used in the sense of "the collection of information of military or political value")
Should I bother keeping an estimate of how many soldiers he has at his disposal or should I abstract that out to stress?
Fate expects things to be abstracted. I wouldn't let that stop me using numbers, but they'd be part of the narrative rather than the mechanics.
A mission (going badly) might end up with something like this:
So, you've split into four teams of 3 to take the different approaches.
Roll Warfare against the Enemy defence. The enemy roll ++++ and succeed with style, creating the aspect "Got them pinned".
The enemy sends some guards around to cover their left flank, Agent Bob radios in "This is squad Delta! They've got us pinned in the West Corridor!". The rest of the attack isn't going a whole lot better and squad Beta is a man down.
On their turn, the enemy press their advantage.
They roll Warfare against the merc's defence, and boosts with the Pinned aspect for a +2 dealing 4 points of stress. That's enough to take out the organisation so the player decides to take a consequence "Squad Delta is hosed.
"Boss! They've got Jenkins and Leroy, and I've taken one in the shoulder, we need to get out of here… now!"
There will come a point where it doesn't make sense for the entire organisation to be wiped out if a mission goes really, really wrong. So when it grows to a certain size you'll probably want to break out an assault team as a separate character. You can then keep modelling it that way (and deal with the bookkeeping) or treat consequences it takes (or if it gets taken out) as stress against the parent organisation.
It's also worth thinking about just how much emphasis you want to place on the organisation rather than the player characters.
The above rather puts the focus on the organisation, but once you get down to the mission on the ground level, you might just want to handle it by treating the organisation as a piece of equipment which gives one or two ablative consequence slots to the heroes (someone got shot? Jenkins took one for the team) and some aspects which can be invoked.