I don't care if it's "God approved" to kill animals to eat them, the way things are right now in slaughterhouses, it's just brutal and horrifying to watch, and the mere idea of killing an animal to eat it just seems bad anyway.

So is this a punishment for original sin and something that wouldn't have happened?

At least you admit that God allowed us to eat meat. That's a good start.

It's hard to say what life would be like without Original Sin. No suffering, having to study, aging, death -- if I could have all that, I'd gladly give up fried chicken.

Even after Original Sin, the world was a VASTLY different place before the flood. There used to be a large water canopy over the whole earth, which blocked most UV radiation and regulated temperature. Imagine no temperature extremes at the North Pole or the equator! It allowed for larger land animals, and the higher pressure/oxygen level allowed everything to be larger, healthier, and more vigorous, including man. And with the increased protection from UV, man used to live to be 700 - 900 years. Plus less of the earth was covered with water, so travel was much easier.

But the world today is what it is. It's very hard to get around eating meat, especially if you do any physical work.

As for the killing of animals for meat, we're not supposed to be a bunch of squeamish SJWs about it. If we lived a more natural lifestyle, not separated from nature and reality, we would grow up around the small-scale, humane harvesting of animals and it wouldn't seem like a big deal.

Also, Disney has perpetuated the notion that deer (for example) are just like us, only with antlers. We've been anthropomorphizing animals utilizing a power (computers, TV, movies) that didn't exist 120 years ago.

Most conservatives (let alone Christians or Catholics) like them a good hamburger -- myself included. There's a reason why Friday (and other) abstinence is a work of penance. It's because meat is so good, wholesome, tasty, and gives a lot of energy and keeps you full for longer than 1 hour.

And I'll share another tidbit: there are some muscular/stocky (no, I don't mean "fat" -- if I meant "fat" I'd say "fat") individuals that really need meat to get enough protein and calories just to maintain their body. Each ounce of muscle on your body requires X calories per hour just to exist -- even more calories per hour if those muscles are used very much. Such individuals laugh at the notion of a few vegetables or a piece of fruit for breakfast.  
And furthermore, there are a bunch of other individuals who work hard physical jobs and thus require more calories/protein and can't be snacking on light, quickly digestable food all day long. And some fall into both groups.

Before the modern age, with hundreds of channels (or millions of channels, like Youtube) there were just two categories of people:

1. Those who butchered their own meat on a farm or worked in a slaughterhouse, and were used to it -- it didn't bother them.
2. The blissfully ignorant -- those who didn't live on a farm OR work in a slaughterhouse.

But here's the catch -- those in group 2 not only didn't know how meat was harvested, but they never did found out. There were no Vegan movements, no "Food, Inc." docuмentary, no Youtube channels, no leftist anti-meat or environmentalist propaganda, none of that.

In the opinion of some, those animals which now are fierce and kill others, would, in that state [of innocence], have been tame, not only in regard to man, but also in regard to other animals. But this is quite unreasonable. For the nature of animals was not changed by man's sin, as if those whose nature now it is to devour the flesh of others, would then have lived on herbs, as the lion and falcon. Nor does Bede's gloss on Genesis 1:30 say that trees and herbs were given as food to all animals and birds, but to some. Thus there would have been a natural antipathy between some animals. They would not, however, on this account have been excepted from the mastership of man: as neither at present are they for that reason excepted from the mastership of God, Whose Providence has ordained all this. Of this Providence man would have been the executor, as appears even now in regard to domestic animals, since fowls are given by men as food to the trained falcon.

I don't care if it's "God approved" to kill animals to eat them, the way things are right now in slaughterhouses, it's just brutal and horrifying to watch, and the mere idea of killing an animal to eat it just seems bad anyway.

So is this a punishment for original sin and something that wouldn't have happened?

Probably by snapping the neck.  I don't have any experience with minks, but with small animals like rabbits, that's the best option.  They usually don't even have time to process what's happening to them, much less suffer because of it.