The only thing wrong with the Enterprise at all during her escape from the Vengeance was her warp drive was not 100% operational due to one coolant feed being sabotaged.
Classically, though, when main power is reduced, the entire performance of the ship suffers.
Vengeance's phasers reacted as though the shields weren't even there at all, not just blowing up but fully vapourising entire sections of the Enterprise's hull
The curious thing, just as you say, is that we never saw what shields are supposed to accomplish in this version of the 2250s-60s. In STXI, after arrival at Vulcan, raised shields do not prevent hull damage from a minor collision; the rest of shield action there is against Nero's reputedly exceptionally penetrating weapons.
Perhaps what we see is normal ship-to-ship combat for the era?
In such a case, one would expect ships to be defended with other things in addition to shields. Close-in active defenses, perhaps? In STXI, Robau and George Kirk tried to use such against Nero's missiles, almost but not quite successfully - and Sulu then demonstrated how easily a modern starship can shoot down missiles that are not coming towards her
. Perhaps that was the key - Pike knew he couldn't defend against weapons coming directly at him, with no wingman to divert Nero's fire, and Kirk in ST:ID had no prayer of shooting down the whatevers that Marcus was firing, but could have defended a third party from these very same weapons?