Is the isolation necessary or just an implementation detail? When my family had a goat we didn't do anything like that, the kids grew freely sucking the mother and then she was giving milk for years after that.
But when we used to own cows & goats, it was very common practice to keep the youngling tied away a few hours before milking. you can feel the pain of separation the parent goes through. ( goat and cow crying, while looking at their offspring is indeed quite sad https://www.flickr.com/photos/murfalizer/4429214117). to add insult to injury it is also common practice to tie the youngling in the line sight of the parent to milk more freely.