Wrong?
Humans are part of nature. You point that out, but I don’t think you take it far enough. Whatever we do affects the so-called “natural balance.” We are part of nature regardless of our actions. If we choose as a species (or subset) to abstain from hunting in favor of farming, ranching, etc., that’s part of the natural balance. Just because we reason doesn’t mean our actions are unnatural.
If we overhunt (killing more healthy adults than arise naturally), there will be fewer of whatever we’re hunting. If we continue overhunting, there will eventually be too few members of that population to sustain it, and an extinction occurs (the Passenger Pigeon being the most commonly-used example), and its environmental/ecological niches will be filled by other species.
If we halt our overhunting, the population hunted will either return to it’s prior state or its niches will have been partially supplanted by other species, and the new stable population will be less than before.
If, OTOH, we underhunt, the species will spread until it reaches a new equilibrium. This may be attained through geographical (Komodo dragons) or climatological (Polar bears) means, or by predation (or myriad other means, of course, nothing in nature happens in a vaccuum). In other words, if a predator species disappears, either the prey will expand its territory until it can’t go any further, or another predator species will take advantage.
The “natural balance” is not static. It is constantly in flux. Predation by humans is a part of the whole, and its degree merely affects the flux. Nothing we can do, short of reducing the planet to cinders, will end the “balance of nature.”
That’s not to say I don’t like the current crop of critters, of course. I oppose the extinction of any species, mostly out of my quixotic opposition to the Second Law of Thermodynamics.
But while I find hunting a species into extinction to be morally repugnant, it ain’t the end of the world.
And neither is letting hunted species go unhunted.