I guess it defines what you mean as “native” as it takes a while for species to hit equilibrium and survive. But environmental effects change that balance as well as humans.
So it seems tautologically that any species in balance is “native.”
In reality, species can easily overpopulate and go extinct or extinct others. See stuff like the North American camel that died out.
Or the gray wolf population in the US West that is growing quite too high, despite being “native.”
In my region, deer are very plentiful in suburban areas because humans crowded out their natural predators. They are native but are overpopulated. Hunting helps reduce their population. If you remove hunting then you must deal with their overpopulation some other way.
The thing is that even with areas where feeding like this doesn't occur like the Mark Twain National Forest, overpopulation is still an issue, so their claim that feeding causes issues is inaccurate, or at least incomplete and more nuanced. It just improves their quality of life and health until they're hunted. Technically it could increase the carrying capacity of the species, but because the population is managed, it's not really an issue.