I have to be careful what I say here because I'll be branded a shocking hypocrite, as I've "released" a herd of fallow deer on my farm (albeit within a fenced area), and I kill them by "hunting" in order to exploit the loophole that enables me to process and sell venison under the same exemptions that apply to wild deer.
Be that as it may, there's no doubt that the release of non-native species purely for the purpose of hunting has caused untold damage to ecosystems around the world, with no tangible benefits. Think of the foxes that were taken abroad by the English gentry in order to be able to hunt in the traditional British fashion on their new estates in the colonies. No benefit, only harm. And still causing damage now.
There is no way that this is comparable with the damage caused by farming, as farming is for food production and therefore has a tangible benefit. And it's closely managed and regulated (or can be) so where environmental damage has been caused steps can potentially be taken to rectify that, as we're slowly but surely seeing happening.
Hunting as a primary source of food production hasn't been a thing of any significance in the western world for a very long time, so any suggestion that food production is justification for releasing species to hunt is laughable. (Ignore my fallow deer at this point please!)
With regard to the release of non-native species in the UK, for supposed "conservation" or "rewilding" purposes, I'm broadly in favour of the reintroduction of species that were present here within relatively recent history (say, since the last ice age), but only where an ecological niche still exists for them. Boar are a good example here (despite not having been deliberately reintroduced) as they've clearly slotted straight back into their place in our ecosystem.
I don't think we've got enough of the right connected habitat remaining intact to be considering the widescale reintroduction of large herbivores (eurasian elk and bison) other than perhaps small domesticated herds kept within enclosed areas, largely for novelty value.
We don't have enough expanse of unpopulated area to be considering wolves.
Beavers I'm OK with (aren't we all?

) with one caveat, which I'll come to in a minute.
So that leaves us with the lynx, which is probably the strongest contender. And I sit on the fence.
My biggest issue with any of these reintroductions is that the species immediately receives blanket protection, which is wrong, imo. The protected status should only apply while they remain within the area designated for their release. Those areas will have been carefully studied, assessments on potential impact will have been carried out, permission will have been obtained, and arrangements will have been made with affected landowners within that area. All well and good.
The proposal to release lynx in Kielder is a prime example. I'm sure lynx would be very happy there, and could have protected status within the designated area. But any animals that stray beyond the boundaries of that area should lose that protection, and be given the status of "non-native invasive species", which is where the hunting comes into it. Non-native invasive species can be dealt with by landowners/managers as they see fit, in order to protect their own interests in the land (livestock, crops etc) and to protect the existing native fauna and flora, provided that the methods used comply with good welfare guidelines.
That is, imo, how it should have been applied to beavers that began to colonise areas outside designated release sites, and it is basically the situation that applies to boar (except there are no designated release sites where they are protected).
(But to get back to the OP's original question about a purely hypothetical situation, I'd gladly introduce muntjac into the wild, for hunting, in this part of north-west Wales, so if anyone has got some please get in touch.....).