I will try (and probably fail) to bring some balance and neutrality to this thread.
The record should show that in the first instance I am against fox hunting with packs of hounds for example but I need to be honest and explain why that is the case. It is mostly to do with the behaviour of the hunt rather than just what they are actually doing. Trespass, damage to private ground, the most unattractive sense of entitlement, blocking and obstructing highways due to this entitlement. The list goes on. None of this can be argued against. It was ever thus and will continue to be the case when hounds are loose to pursue a trail which, with some degree of certainty, will allow them to come across the scent/sight of an actual animal. Every year the hounds come charging through the golf course. There is no public right of way. The huntsmen sit on the road tooting their horn to no avail. The damage is done and grows each time they upset yet another person(s) going about their legal business but having to put up with this unacceptable behaviour. I don't see how their behaviour can be changed and the difficulty grows as they continue to lose access to more land. I know my family have now refused access to the farm due to them not listening to basic instructions about coming in to the yard where the cattle are in sheds. Too much commotion, they get in the way. They think turning up the next day with a bottle of scotch will do. Nope.
But what about the basics of what fox hunting is? Pursuing an animal, regardless of the odds, across land using a number of hounds, men and horses and the noise that follows. It is one or all or some of the following. Bullying. Ineffective. Inefficient. Ignoring the more direct social issues that I take issue with, the basics of what they are doing is not that sensible at first glance.
And a similar concept and mindset applies to Stag hunting.
I would never participate in either. I would not go as far as barbaric because it isn't in my view. But it smacks of superior behaviour, hierarchy suppression, top heavy aggression and more.
I am however always conflicted about the existence of both. I think it is easy for any one of us to say "it is more humane and efficient to put a bullet in to an animal without all the noise and bother and pursuit" And that is entirely correct. One human goes out with a gun, finds a target animal and without much fuss or bother, selects a shot, takes it and the animal is dead. By all measures of being reasonable, it is more effective in terms of cost, effort and impact on all involved in the situation including the targeted animal.
And I feel that is our (and my) mistake when assessing the suitability of hunting with dogs in the modern world we live in. It is too easy to look at it like that. It is more punishing than our ways. It is less effective. It is more time consuming. It is more costly. It pushes the animal to a much more heightened state of distress than would otherwise be the case.
So how is this ok? The question really is not if it is ok but why or how is it reasonable for modern society to pass judgement on our overall evolvement over time. We are clearly animals. Humans have, relatively speaking, been soft and far removed from how we behaved historically for a vanishingly small period of time. Like all other animals we function to achieve certain basics. We need to feed, we need to drink, we need to breed and we need to shelter from certain conditions. We ultimately need to survive and ensure continued existence. In order to do that, we continue to make use of innate feelings, skills and emotions that endure. Mostly those characteristics are now used to build, create and grow but the need to satisfy certain traits still clearly exist for some. In different ways. Chase, capture, kill, succeed. That is not a fault of anyone as such and is enforcing the suppression of naturally occurring traits and feelings fair?
The overwhelming majority of people tend to find initial fault with the thought of killing. I mean Miss Cottis will set mouse traps in her garage but will have to get me to go over and deal with it when they are set off. I on the other hand will happily take a rifle out and shoot whatever is reasonable to shoot within the bounds of legality and what I consider to be proportionate and justified need. In the first instance it is the needing to be done that motivates me to do it. I would rather be playing golf and using skills which to me are more rewarding but when I do have to go and shoot stuff, I really enjoy it. Not just the being outside but the utilisation of skills and equipment to kill. I am not an aggressive person or violent nor someone who has a bad impact upon society. But there is a natural joy to hunting including the kill. It simply satisfies a raw trait that I have and I know that I am not alone. Some might not have it, others might have theirs supressed by decades of society suggesting alternative behaviour would be better. I said I am not aggressive or violent but I kill animals? Am I violent and aggressive? Maybe a bit and in basic terms probably in a similar way to those who hunt using different methods.
So where are we? Is some killing ok? But not others? If I, we, you or anyone else is allowed to enjoy (and if you don't why do you do it?) our form of killing, is it really so unacceptable to understand and accept that others might also need to fulfil the needs they have that are quite natural. Or do we say that society has changed too much? Are we less primitive animals now (certainly we are less primitive) and because of that and fundamental shifts in how society mostly operates, should certain passtimes stop? After all, we have altered our behaviours in lots of ways over relatively short periods of time. Mostly to do with technological advancements but that is really irrelevant other than to demonstrate that our rate of advancements, means that we end up having to constantly self regulate what we do and what most of us find acceptable.
For me, that is easy to do for most things in life. I can see the overall benefit for most people and I think it is reasonable for sacrifices to be made that result in the majority being happy. This subject however always stumps me. I cannot quite reconcile the issues. I end up sitting on the fence thinking "as long as nobody is directly harmed by behaviours" then I am ok with people satisfying natural traits. It's hard. It is a coin flip with me hoping it comes down a certain way

. I cannot figure out if I am a hypocrite or if my ability to be subjective is flawed and compromised due to me knowing I receive some element of fulfilment from killing animals. Yes it seems reasonable and justified but it is still killing and I am very comfortable with that.
But to try and answer your question. Yes, I think a single reason is that this pursuit that most of us find so disproportionately unreasonable, is satisfying very natural, normal and emotional urges that animals have and in this instance those animals are humans. Those who feel it understand it. The rest of us don't. The basis for all debates basically. I don't think it can be classified as a wrong. It is however deeply unattractive to many people and I can understand why. That is why I would never be associated with any of it.
I sort of answered your question without really answering it. Or I suspect we are not wholly comfortable with the answer. In more ways than one.