Well, its kinda like how dogs can be treated with a lot of cancer drugs people can’t- its largely a matter of longevity and personhood. Just load 'em up with stuff that’ll cause all kinds of crazy side effects in a decade or so, if they’re at least six, even in longer lived breeds. (assuming its a bad form of cancer in the first place)

Sheep don’t have to worry about the things people have to worry about when they aren’t going to have to use the same body for the better part of a century.
So yes, they do get sick, but its not regarded as the same level of devastation because its an animal, you can use harsher treatments, and you’re not looking at the same level of magnitude of suffering or life lost.

We also work a lot harder to keep a lot more physically frail individuals alive whom would otherwise just be dead if they were sheep. For example, I had a doting mother who got me medical attention for my frequent dehydration as a child, and I am now able to advocate for myself when I have those episodes.
A sheep who couldn’t tolerate something foul on some grass would have likely died as a lamb and have just been a sad statistic before they even developed enough to graze.

If we had much shorter lifespans, we’d honestly be some pretty rugged creatures. Our stomachs are only .5 less acidic than some carrion birds. More acidic than most carnivores.
No worries about preserving teeth or joints, eat what you want, its literally Yolo!

Our brains, vision, bipedalism and weird vomiting thing are pretty damn sensitive though.
A lot of animals don’t have such a vomit issue. Rats literally can’t vomit, iirc.