I'm on the fence about this.
On one hand I agree with Charmmy. While I have no problems using contraceptives (and I do), I also do think inbreeding is at an extreme level right now. It's an issue I had since beta. I'll take their example of flamingoes because that's a really good one. There's zero reason for relatives to be mating if there's other available mates for them. Do you really think zoos irl would be able to manage inbreeding in birds like that, when they have flocks in the hundreds?
Cases like that, and other animals like herd animals, should be less receptive to breed with one another. Of course it should only be 100% safe if you use contraceptives but if a flamingo has a mate they shouldn't just suddenly change to mating with their child, and vice versa. The children should only mate with relatives if there's no other options (so again, use contraceptives to be 100% safe).
On the other hand, and I'm sorry I don't mean to offend but, I disagree wholeheartedly with disabling inbreeding altogether. You can't really take a human emotion and apply it to a situation like this, and then try to make a moral argument out of it. I'm sorry that it disturbs you so much for animals to inbreed, but it's just a way of life. Animals don't have morals like we do, so their idea of mating with parents/siblings/children doesn't affect them the way it does with humans. If it bothers you, like others have said, figure out ways around it like in real life. For example if your dog or cat were to have puppies or kittens, your work around is to get them neutered/spayed as soon as they're old enough to prevent them from mating with one another. Similar idea here.