Cooldowns can be a drag at times, too, while they're here to stay with Underrail, my point still stands on how you could try to design a similar game without cooldowns.
There's multiple problems with what you've just said:
combat consumables would be quite limited instead of having cooldowns, restoratives, grenades, etc (non-combat healing is still easy enough to acquire in emergencies)
That's how it works in AoD. But AoD is a much smaller game and easier to balance; Underrail is huge and not only that, but a lot of content is randomized as well. It would be nightmare to properly balance the economy that way.
there would be a single resource that all the special abilities of all kinds feed from, with an associated stat
That's no better than a cooldown system
equipment focused abilities that drain item durability or even break the items
Don't know about you, but I hate durability concept in games even more than cooldowns
the "uber" character abilities with 10 turn cooldowns would just exhaust or debuff the character in some way
And you will lose all incentive to use them. Imagine LoC with debuffs
abilities with 1 turn cooldowns that are balanced around being used only once per turn because otherwise they'd be very strong (Point Shot) would be marked as "turn ender" actions to avoid the cooldown, meaning they can only be used as the final action that ends a turn
Ironically, this makes system more rigid instead and limits your choices more than cooldowns - while being exactly as much an arbitrary limitation
Also,
S0rcererV1ct0r, your idea is to replace cooldowns with critical failures. There are some problems with that as well. You see, cooldowns are deterministic. As you use the CD-ed ability and wait for cooldown to pass, it still allows you to plan - you know when the cooldown will end and can plan accordingly in the meantime. Critical failures are pure RNG, you can't plan around them, you can only have the back-up or retreat plan. Even in Fallout, Critical Failures were used just to spice up the combat, not to limit the player.