Let's see, this is gonna be a long one:
Replacing Energy with Cooldowns sounds pretty interesting, it prevents one from spamming the same ability over and over again to get through encounters. Energy is useless for preventing this, because enemies aren't strong enough to survive through 6 straight turns of 300% damage. Queen of the Deep comes close, though. As the energy system is now, it just serves to have Tech Specialists hit the "Rest" button after every 2-3 encounters, which gets really annoying and doesn't do anything to limit the other 2 classes because they have built in energy regens and no enemy can survive long enough to expend their energy reserves. Smugglers in particular only have to worry about their Flashbang landing (which is pretty easy as it is now) and boom! 160 damage a turn.
Gyms, Syri, and Tanis stat boosts should definitely be changed to not give stats indefinitely. There isn't much reason to put any thought behind your build right now besdies maxing out Willpower because all the other stats you can get later without using up Stat points. However, skills would have to be heavily replaced/retooled for this to happen. Because we have the Grenade skills that deal damage based on Int and if you don't train Int, they just wind up being useless. And you have no choice but to take it. Meaning that character builds wind up being very linear. Smugglers won't take it because they can already deal ~160 damage a turn if Flashbang works. So people who make Smugglers just wind up leveling Aim and Reflexes. Physical Mercenaries could take it because their damage and means of breaking out of binds is in their Physique. But Ranged Mercenaries can't because no matter what, they have to level Physique to break out of grapples, so one build gets full use out of a Skill and for another build, it's a complete waste of time.
Passive abilities being the majority... I'm not sure about this one. If done right, you can appeal to those that just want to mash the Shoot button and fap, but if done incorrectly, it could wind up being like the level 8 perk, in which it's just a "do you attack from a distance or up close?" checkbox. It could make the game shallower than it is currently if handled incorrectly. I'm not saying this is a bad idea, just that you should tread carefully if this is going to be the goal.
The resting mechanics as you've laid them out make sense to me, if energy is taken out. Enemies being more likely to attack you while you rest for four hours isn't totally insane. Healing half your health after battle prevents you from having to rest every 2 or 3 encounters like the energy system does currently.
Base modifiers are pretty straight forward, I have no real objection to it. Makes equipment more personalized too, which is always a good thing.
Exhaustive abilities... I'm not too keen on those. It's already rather easy to trivialize bosses as it is now, abilities like that would probably allow for even more game imbalance. Abilities as overpowering as that shouldn't exist on bosses and they're utterly pointless on normal mobs.
For your ability advancement scheme... I'd have to see the actual abilities to make a call on that. Your actual structure is pretty... structured.
And I 100% agree with your Specialization idea. Balanced properly, that allows for a hell of a lot build freedom than if the current system was balanced. Being able to take the specific abilities that you want would make for very interesting build possibilities and people would probably start taking builds more seriously.