The first thing that occurred to me was kobe-breed beef in Japan being fed fine, human-grade grains and massaged with sake by beautiful women. While I know I'd probably enjoy that, I think we need to think more about the animal perspective than the human.
For anyone who's had a dog made to feel uneasy by some environmental trigger, fireworks or thunderstorms come to mind, it should be obvious that even a creature who's lived so closely with humans for so long in their development isn't comforted by the same things that comforts their humans. Giving a dog a hug and pats doesn't really make them feel any better about the sky exploding. In fact, their human being so worried about their canine being so distraught is bound to make things worse. Aside from a sound-proofed space, I don't think anything could make a noise-averse dog feel better in those circumstances, but for the one who confidently commands and rewards to suddenly be worried and looking to the canine for cues is strange behaviour. Dogs like routine and the normalcy that follows it, and the predictability of their human acting as their human does.
Most of these animals are prey for something in the wild, which means that their sense of safety is informed by that predator fear. For some that means plenty of cover that also function stacks as some of their forages of choice. For some, that means really good sight lines, and the ability to herd together for safety.
It also means more subtle things, in my opinion. Let's take rabbits, for example. They are prey animals. For that reason, they generally don't want to be picked up, because if a predator were to do that to them, it would mean imminent death. It is so ingrained in them that if you flip a rabbit upside down, which I entreat you all not to do, except at extreme need, they go into a state of shock-driven paralysis, which can actually cause heart attacks in weaker individuals. So for rabbits, one of the things we can try to do is have a system that emphasizes that rabbits are not handled in that manner, except at extreme need.
Now I am not saying don't pet rabbits, and don't get cuddly with them. I'm saying that to properly show them affection, it has to be on their terms. It is necessary, because we are such large creatures that smell like omnivores, to get down on their level in order for them to get to know and trust us. If we pick them up and insist they sit on our laps, some part of them, even if they sit still for it, will be certain that hungry jaws are soon to follow. If we get down, on the ground, go nose-to-nose, and massage their cheeks and the tops of their heads, from the nose, between the eyes, to behind and between the ears, well they'll see that we most certainly aren't rabbits. But they have all four on the floor, they can escape any time they choose, and from their perspective, the finger caresses they are receiving are really close to the way rabbits are groomed by their herdmates.
But to treat such an animal as though it were a dog or a cat isn't pampering that animal, even though a cat or a dog might consider it as such. They aren't comforted by the same things, and their anxiety isn't triggered the same way either.
So to truly pamper all of our animals, I think it's necessary to take onesself out of one's human perspective (and even for vegans, it's helpful to assume that they view you as an omnivore), and try to view their environment and interactions with humans as potential prey animals would view their environment and interactions with a potential predator.
I definitely agree that making sure they have endless clean drinking water and a constant source of food that is safe to eat as much of as they like (hay for rabbits, for instance, as opposed to a more nutrient-dense pellet, that can cause bathroom problems and/or overfed conditions), and a variety of foods that they like to supplement with, are steps that can keep animals from getting sick, or resorting to behaviour that might injure them, like eating medicinals because their regular feed is contaminated.
But for it to be a proper pamper, it can't be just the kobe beef pampering, a human-centric judgement. It needs to be an [insert specific animal here]-centric assessment of how they see their existence, which is a lot more difficult than assuring they have proper feed, shelter, and room to move.
Though don't discount the massage thing. I have found that my rabbit loves full-bunny massages. She actually gets tiny bunny knots, and will binky around after I work them out. It's just that pigs will probably prefer to be massaged with mud or pig shit, for instance.
One other thing that occurs to me is insects. If there was something that could be used as a scent deterrent against biting insects, something herbal, it would probably result in less animal stress. If this were a spray that could be applied to each animal as it exits its shelter, semi-regularly and certainly after every rain or when insects seem to be bothering them most, it might result in more time grazing/foraging and less time tail-flicking and running away from clouds of insects. It might, in some animals, also cut down on insect-borne parasites or infection of insect wounds.
I would also explore herbal calmatives, at least insofar as ensuring that those herbs that are safe for that purpose are present in their paddocks' animal medicine chest. If chamomile, or any other herb that produces the desired effect without side-effects, is safe for them, I would make sure there's lots of it available. If they need a calmative, they can treat themselves to it. I don't think cows can get stoned on chamomile, but I know that my rabbit sure loves the taste. I suppose the greater observation here would be to make sure they have a fully stocked animal apothecary in addition to way more optimal forage than they could ever run out of, so they don't try eating medicinals as roughage.
These are just a few thoughts. I would love to see other ideas. It's an important sphere within the realm of permaculture that could really be done well by those that care.