More a key role in tourism and managing deforestation. Natural resource management agencies take care of the difference at the end of the season to prevent starvation regardless of hunter involvement.

We almost hunted them to extinction and our continuous destruction of their habitat and eradication of their predators means they can basically never reach equalibrium again, so it's just a constant risk of over consumption, over population and starvation.

Hunters mostly make a lot of money for the area selling the license, both through fees and the economic activity of the hunters.
Actually letting things get better would involve reintroducing a non-trivial number of wolves, which is largely opposed by farmers and some hunting groups since it would reduce the population of deer.

So they do currently play a key role, but largely because it's something they want and it's generally pretty profitable. It's just treading water though, since no one with power is particularly interested in fixing it.