I agree with you that much of the time we can use abstracted stat blocks in our games (regardless of what we envision is behind them!) Caveating that for me an absolute resistance to NPCs with character classes feels less helpful. Say our PC Monk has the common backstory of studying for years at a monastery containing 144 monks. None of whom are allowed Monk character class levels!? As an aside, a flat-out disconnect between PCs and all other inhabitants of their world causes dissonance... for me at least!
Can we directly equate CR to tiers? You seem to be saying it is 1:1 to class level. Is that right?
I'm not against statting the occasional NPC out with the PC rules. Depends on the character. It's workable in 5e (unlike 4e).
PC CR is probably somewhere around 3/4 of their level in 5e. 3e tried Level = CR and it didn't work. Pathfinder went with Level-1 = CR, doesn't really work either but closer since 3e/PF CR system is +2 Levels = x2 Power.