I think the problem lies in fact that there is no ability that is demonstrably unique to one class and one class alone. Either an ability cannot be quantified measurably (aka: only fighters get action surge and people in-world can point out action surges), the ability is shared among other classes (aka fireball on a dozen spell lists) or you can just make an NPC with that class ability and no levels in any "class". (Personally, I find the last one cheating, tbh). Ergo, you cannot scientifically measure a class by its class abilities: even if you put Bob, a "paladin" in a room and say "smite that orc" and he does, you can't say use it as proof of paladinhood existing because a.) We'd have to trust the action he did was "smiting" and not something else, like a crit or spell b.) We'd have to assume Bob actually had levels in Paladin and not some other class that grants Smite as an action and c.) Bob wasn't just a totally custom NPC with 5 HD and Smite as a Action but has nothing else to do with paladins.
Of course, the latter DOES raise the question of "If Bob is a totally unique NPC who just somehow gained/learned the Smite ability without taking any levels of Paladin, why can't my PC do the same thing?" Or "If Joe the Archmage is using a spellbook to cast rituals but isn't a wizard, can Joe teach my Sorcerer how to do that?" That can have one of two effects: it removes ALL fluff from classes (sure, you can learn smite, but you need to take a level from the class commonly called a Paladin except you're not actually becoming a paladin, its just the name of the metagame concept that grants smite.) or it begs for a more granular method of divvying up powers (taking class abilities with feats, swapping class features from one class with another, using talent/feat trees and large generic classes, or buying abilities ala cart without need of classes). Or there's options three: (Bob learned it because he's an NPC and not bound to the laws and fluff that govern PCs, but you are forced into one of 12 predetermined powersets with a few choice points).
