This is accurate. Goats and cows will literally beg you to milk them if you're late doing so. Chickens leave most of their eggs to rot, only incubating very few. Isn't it more disrespectful to the animal to let the food they provide go to waste?
Genuine question: Why do the animals beg you to milk them if they presumably have children? The animals only produce milk after having given birth, right? Are you saying that the animals produce so much milk that they still need to beg to be milked after their young have been nourished?
Or are you saying they only beg for milk in situations in which their young have been taken away?
Cows and Goats will both wean their offspring on their own, and continue to produce milk for years afterwards. Milking them keeps them from getting pregnant again too soon.
Is this true? Everything I'm reading online says that the animals will naturally "dry up" after the young have weened, and it's only by humans manually prolonging the lactation period that animals are able to give milk long after having given birth.
Yes, if you don't milk them after they wean their young, they can dry up. However, modern dairy farms let cows choose when and if they're milked, and the cows choose to be milked twice a day, and seem to really enjoy it.
We should also separate cows from sheep and goats. Where I am (prefer not to disclose exactly) sheep only produce milk between January and September and goats between April and November, the periods that coincide with their birth and weaning periods. And there's only few cows here because of the geography and the climate.
I think that in other places sheep can be milked year-round by having them breed twice in a year, but I'd guess that this would tire the animals and reduce the quality of the milk and probably also the quantity (which is already quite low for sheep and goats). I really doubt goats can be convinced to do things the humans' way. I've never heard of anyone industrially breeding them and I can't imagine goats nicely lining up to be milked by a robot!
A lot of the lore that animal rights folks share about how dairy animals are bred and milked seems to be collected from a very particular kind of practice, focused exclusively on dairy cows in the US, and that isn't as widespread as the animal rights people would have you think. For example, I bet everyone has heard by now how cows are cruelly separated from their calves immediately when the calf is born, but I doubt as many people have heard of Salers cows that will simply not produce any milk unless their calf is touching them. Milk from Salers cows is a legal requirement for some PDO cheeses so while their numbers are few compared to Freisians, Jerseys, etc, they are no less cows and no less dairy cows. They're just the inconvenient kind (for milking, as well as for propagandising).
