Yet even in TOS, you got plenty of variation among individuals. Kang was kinda the prototype for the gruff, honorable warrior popularized by later Treks, whereas the guy in "Friday's Child" was a cowardly weasel. Kor (as seen on TOS) was a cynical pragmatist who saw the occasional massacre as an unfortunate necessity when it came to imposing order on occupied territories. Koloth was an urbane smoothie who preferred spies and sabotage to combat.
As for TNG, I don't recall "Heart of Glory" well, but that was just a handful of Klingons, right? Wouldn't be hard to find plenty of groups and communities of religious fanatics in most city in the world today. Maybe that was just a particularly fanatical batch of Klingons?
From where I'm sitting now, there's a Jewish temple on the corner and a Baptist church one block over. And that's just in one neighborhood in one town in one state in one country on one planet. Have to assume that the Klingon Empire is at least as diverse as, say, Brooklyn.