There are definitely more Klingon professions shown on screen. There’s a Klingon Chef working on DS9, there are more than one case where we see Klingon lawyers and I remember at least one Klingon Scientist. And while B’Elanna is only half-Klingon, her character shows how a Klingon could work as an engineer. All of them probably cases of toxic work environments, but they seem to work. What we see mostly on screen are the crew of warships. Other species must think every Human is a scientist or diplomat.

simply no way they would ever have adopted the scientific method let alone made it into space.

Isn’t it canon that they stole most of the spacefaring technology from the Romulans when they made first contact?

That was especially Ezri Dax and IMO a great way for the writers to show that Jadzia and Ezri are very different. Jadzia was very into Klingon culture and kind of celebrated the backstabbing and cheating way of the Klingons. She played the game and enjoyed it. Ezri was more like “you guys are hypocrites and you suck”