The truth is that the exact origins of noble families are unknown. There are two main hypotheses, one of which posits that they emerged from the various local warlords that held territory with their militias (whether that makes the leaders themselves warriors or not is up for debate). The second one says they came from the families of public servants in the Roman Empire, which were often given hereditary, honorable titles. In that case they would almost certainly never have fought, being exempt from any kind of military service. The truth is probably a mixture of both, though in what proportions, nobody knows.Nobles used to be powerful warriors even on Earth.
But take a look at the very party that killed said Demon King - was anyone elevated in station beyond just being celebrated? The FMC was noble from birth, Raven remains guard captain, the hero is a celebrity for the peasants and the mage just follows him around. It was mentioned earlier that each country names their own hero, so you're right that they're interested in monopolyzing said individuals, but it doesn't seem like it's a "slay the dragon, get the princess" kind of deal. It does probably come with financial incentives, so they're essentially mercenaries in the service to the crown. So no, it can;t be said that this is how it works in this manga. Whether or not it should, by pure logic is another matter. Perhaps these people are powerful, but not nearly enough to actually threaten a whole kingdom by themselves.But, it's absolutely how society would work in this kind of world. Why else did they send a "party" after the Demon King and not armies? Guys like this would be nukes, they'd be scouted and tied to a country as a form of political power. Obviously it's just a lighthearted story that adapts modern sensibilities to Japanese fantasy realm tropes, but if you want to bring in logic, that's what you'd get.