Maybe we don't see subject world natives fighting for the Klingons because the Klingons would rather be the ones doing the fighting to begin with.
The Klingons may glorify war, but they can't be completely irrational or impractical in their policies or they'd never be able to sustain a functional state. As a rule, the majority of the population of an empire would consist of members of subject populations, and it would be militarily foolhardy to refuse to employ that vast source of manpower, especially given what a manpower-intensive process combat is. If only the ruling culture went into battle, they'd end up getting killed off and there'd be too few of them left to keep ruling over the others. The empire would fall apart.
Besides, there have been plenty of human empires whose ruling cultures have glorified combat as much as the Klingons did -- the Romans being a prime example. And the Romans employed
auxiliary troops from their conquered provinces because it wouldn't have been practical for them to do it all themselves. For that matter, the British had a similar sense of machismo and honor, a dueling tradition and all that, but they routinely employed subject peoples in India and Africa as cannon fodder.
And if it's the glory of combat you want, well, it's the commanders, the officers, who get the glory while the grunts are doing the bulk of the dying. Sure, in principle, Klingons consider dying in battle to be their highest ideal, but not all Klingons would be equally true to their ideals, especially when there are practical advantages to a commander staying alive and sending provincial grunts out to die in his stead, rather than getting himself killed and squandering his experience. A lot of the Klingon bluster about embracing death has got to be a load of bull. Combatants who are trying to die aren't very effective in the long run. The priority in combat is to stay alive. The commanders just talk about the glory of dying in order to convince their cannon fodder to accept being treated as disposable. But the commanders themselves probably have no intention of dying if they can avoid it. Even if they personally have no fear of death, they would have a responsibility to stay alive and protect the investment that the state put into their training and experience.