D&D 5E What if we got rid of stats entirely?

[Umbran]

This would mean all members of a class of a certain level are all equally proficient in a given activity and I find that wildly implausible.

In 5e, NPCs don't typically use the same class definitions. PC class is a concession to our playing a game, not a setting reality.

Your PC is unlikely to interact with sufficient individuals of the same class and level to notice, statistically speaking.