Animal limits do not work automatically. It is necessary that the villagers kill the surplus animals, and for this they need time available.

You did not put screenshots of your village, but in the video (4:17) I can see that the player has placed a limit of 55 animals in a population of 60 villagers.
In addition, he built 5 or 6 weavers and even put the fields too far from the village (though there is free space next to the village), unnecessarily increasing the workload.

With high workoad, the villagers can not kill the animals and the village literally ends up drowning in goats, pigs and sheep.

The usual limit for animals is one animal for every two villagers.