Even though it doesn't exist on screen, there would have to be more nuance to this society than what turns up on tv, or indeed in the books, if it were to be a ... realistic ... society. Surely there must be many different forms of this society, many layers and different aptitudes - not simply the one-uniform wearing kind of monosociety we are most familiar with? Also surely our view of the society is skewed by seeing through its aristocracy, whose martial believes might be very different from other classes, and the depiction of which cannot genuinely reflect values and cultures from different worlds - each with hundreds of years of linked-yet-separate development.
Not all Klingons are warriors, just the majority of the ones we see on screen. In "Playing God" Dax can be seen flirting with a fat Klingon cook with an opera voice. In "Rules of Engagement" we see that Klingon lawyer who was actually a pretty cool and provacative character. And in one of Kor's DS9 episodes, he complains that young people are now opening resturants and opera halls among the aliens he killed in battle when he was young.