Well, that’s the counterargument: “See, they have personalities, and…opera.” But watch with a more jaundiced eye (my intellectual liver has been strained by years of pop culture) at how they are treated by the writers reveals a lot less egalitarianism that it seems at first. TNG-era Klingons are always going on about killing each other for pride, the glory of dying in battle, et cetera; your basic barbarian cultural characteristics, with the humans in attendance watching with kind of a bemused tolerance at such unenlightened antics. On DS9 they’re often called out for breaking Quark’s holosuites and doing other casually wanton acts of damage, and there is a running joke about how Worf’s love-making is so aggressive that Dax has to seek medical attention on a daily basis. (Being familiar with Klingon customs, she apparently gives as good as she gets, but still, it harkens to an image of a race of savages that have to be tamed to mingle with civilized company which is recognizable to anyone familiar with the stereotypes of “the negro” in Western culture.)
And yet, they aren’t some Bronze Age culture uplifted to starfaring state or “primal man uncorrupted by modern civilization”; they’ve built an interstellar empire, developed warp drive and directed energy weapons, transporters, and the other technomagical gadgetry of the Star Trek universe all on their own to the point that they are the military and exploratory equal of the vaunted Starfleet. Somewhere behind the curtain of their warrior culture, they have engineers and scientists and explorers and program managers and accountants…but according to what is presented about Klingons in the context of the show, those would all be dishonorable professions whose practitioners should be fair game for abuse and arbitrary killing. Klingons are not orcs in the sense of being mindless killing machines but they are often treated in orc-like fashion by the writers for the sake of creating an opponent that can be dispatched (in oddly bloodless fashion despite the fact that they mostly use stabby weapons) without any moral consequences, and who will fight to the death with each other over minor offenses.
Sure, you can find instances where the writers subvert this superficial trope with some kind of Klingon counterculture (see, Worf drinks prune juice and calls it a “warrior’s drink!”) but that’s kind of like painting a Ferrari grey to make it less ostentatious. Michael Dorn–who is a fine and much underused actor–tries his best to imbue Mr. Worf with a real personality beyond just being the example of how dangerous a situation is by running headlong into danger and being flung aside by the threat-of-the-week, but he’s working against this established stereotype to do so, and the general impression given is that he is always struggling to contain “the beast within”…which is very much a European colonialist view of “taming the savage races”. You can see how that leaves a bitter aftertaste, even if the Klingons are fun in a boisterous, great-fun-as-long-as-they-aren’t-stabbing-you way.
It is curious that you note the translating of Shakespeare to Klingon, because the original reference for this is Star Trek VI: The Undiscovered Country, which is the one bit of Star Trek canon that unambiguously calls out this stereotyping head on. The uncomfortable diplomatic dinner where the Klingons end up calling out the Enterprise command crew for their implicit (and in Kirk’s case, explicit) bigotry is a stark contrast to the near-constant smugness of the Federation as being otherwise morally superior to Klingons, Romulans, Ferengi, Cardassians, et cetera. It highlights how patronizing it is to treat the Klingons as a singular archetype, which is what every other Star Trek movie and show does on a regular basis.
Of course, the reality is that no alien culture would look or act anything like humanity, and aliens are widely portrayed as humanoid monocultures in television and movies both for the convenience of production (e.g. you can make a new alien race with a forehead prosthetic and a few colloquialisms) and because human-like aliens are far more relatable than some kind of intelligent motile fungi or a post-singularity society that has transformed itself into a cosmic megastructure. But to make it relatable writers and producers often resort to using recognizable archetypes, and in the case of the Klingons, that archetype is of the brutish subhuman just capable of controlling themselves sufficient to wear clothing and make polite noises. And that is basically a classic fantasy orc of your AD&D variety, suitable for taming, enslaving, or murdering as you see fit.
Stranger