Well, the main problem for me in general for animals - it is the less time you giving to them and the game at all - the more time you will require to kill low q ones, sorting, butchering, etc.
So when you close to drop the game, and starting playing not so mutch, game starts require even more time for animals that breed without limits.
I offer next mechanic:
1. Add animal diseases, that can block breeding, decrease animal health or even kill it if will not treated.
2. Increase animal breeding a little to give a balance.
3. Diseases chances greater if animals closer to each other
If this mechanics will be balanced well:
1. Animal will breed fast if you will pay attention to them.
2. Animal caretaking will require some meds
3. The more animals you take - the more attention you will require for them.
4. Overpopulation make it harder to breed more.
5. When you do not pay attention to them - they will breed very slow.