Condense Milker, Gelder, Sheerer, Animal Hauling, and the currently useless Animal Caretaker into a single skill and handle their particulars with knowledge checks.
While the diversity of skills makes sense, and no amount of milking will ever make gelding obvious, it's kind of unusual that a farming society wouldn't have more general knowledge of all of the animals they raised and used for industry. If a society has managed a domestic animal and industries resulting from its byproducts any Dwarf capable of working with that animal at all should have an understanding of how to milk, sheer, geld, collect its eggs, and care for it by the time they're adult farmers (potentially even butcher it, but needing to be familiar with an animal to do that could destroy the meat industry right now). A single Animal Husbandry skill could serve for the skill check, since none of the individual skills has quality modifiers anyway, and the particulars would be required in the same way knowledge of poems are. A society/Dwarf would need to know sheering in general, or have a way to learn it, but once they're familiar with the concept and the animal (this could go by how trained an animal is, or the civilization's familiarity with a species, or eventually an individual check if civs start springing up based on adventurers with a particular knowledge set) the whole interaction of moving it from a pasture, getting it into place to sheer, and sheering it should just be one skill. Unfamiliar animals should be harder to move or more likely to bolt instead of following, and should result in more failed milkings and sheerings until more familiarity is gained. This could translate to an increased rate of domestication if your Dwarves are raising a sustained population of something that you don't have a lot of opportunity to train.
Condensing the skills removes some clutter in the labor menu or its eventual successor, leads smoothly into more knowledge based checks for skills, and makes some Dwarves less worthless. With animal hauling attached to a skill, and having that shared with other essential farming skills, a whole group of peasants suddenly has a lot of new interactions available to them. Every time animals need to be led somewhere, whether they're pulled or herded it could require the Animal Husbandry skill, and subsequently lead to quicker milking, sheering, and gelding just from being more comfortable working with animals (or that particular species, in theory). Moving animals should already require skill checks, since it's not really obvious to someone that hasn't raised a particular animal how to get it to move when it's got several hundred pounds on them. This will make it more valuable to have a whole crowd trained in the skill, since you usually end up with a glut of these skills already, but could use numbers like that when you need to get your herd inside before a goblin raid.
Animal Husbandry could have negative quality results once those are implemented. Animals being guided could run loose, or ones being gelded or milked might attack. Once Dwarves attain a certain level of ability this should be more of a speed check, and assume a legendary Animal Handler or whatever is just going to know how to guide that sheep in any given instance, and at the animal's top speed perhaps.
All of the subsequent parts of the animal industries would remain on the same skills, because there can definitely be qualities to cheese and thread in ways that don't translate as well to gelding.