I get the impression there are Klingon engineers and doctors but they are ridiculed and considered failures. Kind of like nerdy kids in southern states. Also I wouldn't be surprised if they repurposed conquered peoples as engineers and doctors.
What I really like about this episode is the way Kurn tested Worf's klingon-ness. Being a hard-ass for everybody but coddling him to see if it would piss him off.
And I also like it in Ron Moore episodes how he portrays Klingon honor as something hugely hypocritical. Like Worf is the only Klingon in the universe who really acts honorably and politicians are just as devious and opportunistic as Earth politicians except even more murdery. Worf longs for an idealized version of Klingon honor that never really existed and keeps trying to vainly intervene to push the empire in that direction. When really the people who talk the most about honor are only invoking the term to stay in power and are really just doing what's in their own best interest (Also just like Earth politicians).
Honor is the Klingon version of 'family values'.