Whether their warriror class respects other classes is not part of the equation. Others have to do what they can to make a living.
I think it does matter. If Klingons believe being warriors is the only worthwhile endeavor, then how many Klingons will follow that path despite it not being where their talents lie? How many Klingons died in battle with supposedly inferior forces because those warrirors were really better suited to careers in science, engineering, law, or finance?