You're not soppy. There's a bunch of stuff to unpack here - much of which reasonable people can disagree about - but I'll tell you where I'm sitting (which is not without contradiction or cognitive dissonance). I'm definitely not trying to persuade anyone here, and I completely respect your stance.
I eat meat. I could become a vegan (and deeply considered it) but after a lot of reading, listening and thinking I came to feel that while it may be a method of harm reduction, it doesn't remove me from suffering inflicted on other sentient beings (Robert C Jones is a vegan philosopher and interesting voice on this subject). I could drone on about this (for example, the wheat crop in western Australia is responsible for the poisoning of over 1 billion mice per annum) but let's just say being vegan doesn't mean no dead animals. For some people this is the best way to reduce harm,and I both appreciate and respect that choice.
Given that I eat meat, I try to be conscious of where it comes from. The majority of my family's animal protein comes from my in-laws hobby farm where I see the cows range, graze and swim on an acreage run under principles of regenerative agriculture. No tilling of soil (which kills a lot of animals), no packing animals in - but at the end of the year they are sent to slaughter. While the animals are (by any western standard) humanely slaughtered, it is worth reading into what those standards are. Let's just say the majority might go clean, but a significant minority to not. This makes it easier for me to live with than supermarket / factory farmed meat, but I'm still not completely comfortable.
Which brings me on to wild game. These are animals that live longer and freer than domestic livestock. If they are killed by a hunter that is a life cut short, but for wild animals there are no retirement homes. Death is almost certainly coming from either famine, disease or predation, and none of those are quick and merciful (do yourself a favour and don't learn about how wolves feed in elk and moose calves). There is no joy in the kill for many hunters - I wouldn't go as far as to call it regret, but a sorrow and a very strong connection to the tragedy of the act is a lot more common than I think many non-hunters would perceive. Most people cry on their first hunt, and many continue to have a great deal of internal conflict from then on. There is often a feeling a triumph in that they may have been tracking that animal for weeks or months (hunting in most cases is actually incredibly difficult), but I do not think that the majority of hunters have a blood lust. Obviously there are pricks in every community, but in my experience they are a small percentage.
So why use guns? It's a fair question. For literally all of the hunters I've spent time with, an ethical (quick and clean) kill is paramount. A 30 calibre bullet to the lungs of anything outside of Africa is going to cause death before the animal feels much more than the shock of being hit.
Let's go down the road of levelling the playing field some more though.
Bow hunting is popular, but compound bows (with cams to increase the energy applied to the arrow) are often used - again in the name of making a quick kill, so there a technical advantage there. Recurve bows (traditional bow - think Hunger Games) transfer much less energy to the arrow so are a much trickier tool to use consistently if the aim is a kill shot. In either case, it's fairly typical that the animal will run after being shot, so you're tracking a blood trail rather than watching an animal drop. There's a reason bows aren't used in slaughter houses, though this is still a much quicker death than wild animals typically experience.
If you really want to see a human spear, stab or wrestle a deer (ungulates being by far the most common source of big game meat) in order to kill it and then eat it, well at that point I have to say I just disagree - I think that's cruel to the animal. Bear hunting is another subject we can chat about if you want to - I leave that for now though. We do have pumas in North America (and there is limited hunting), but no tigers so I can't speak to that.
So for me confronting and bring directly involved in the tragedy of taking life to sustain life is somehow cathartic to me - I recognize that no human on earth exists without the death of other sentient beings, and this is the way that right now (and I'm open to change) brings me the most peace in terms of minimizing that suffering.