Forest Nymph, the main reason to let animals that have co-evolved with us and like us die out, say dogs, cats and domestic rabbits, is their vulnerabilty. For example, did you know it is estimated that 10-30% of the leather sold in the US is from the black market dog and cat leather industry? In small pieces, dog or cat leather is indistinguishable from cow leather. The only way to know for sure is to do a DNA test on each piece coming into the country. The reason this dog or cat leather is produced is that it is so much cheaper to raise dogs and cats for slaughter compared to cows. Obviously the dogs and cats are treated horribly, crammed into cages together so tightly they can't move, and then are beaten or skinned alive, etc. They don't do things like that with wild animals, like the wolf or african wildcat, or even the dingo. They do it with easily tamed animals who have a long history with us, like dogs, cats, the domesticated fox, the domesticated mink, etc.
Same thing with rabbits. Right here in the US, there are rabbitries everywhere. People breeding rabbits for meat, leather and the pet industry. Even the ones who wind up pets have horrible fates because most will live in tiny cages by themselves their entire lives, but they need a lot more than that to be happy. It is the docile nature of domestic rabbits that makes them suitable for breeding by these people.
You say as long as people aren't breeding them for profit, it's OK to allow them to breed if it is a choice between that and letting the species die out. So let's say this happens: you allow the last two cats on earth to breed, perpetuating this easily tamed species. A hundred years after this decision, there are hundreds of people breeding cats, all kinds of torture breeds with flat faces and long hair. The cats' faces are so flat they can barely even breath, eat or drink and their noses are always running. And people are hoarding cats and all their cats have FIV, which is the feline version of HIV and they all have fleas, ticks and matted hair and are living in their own feces and have no clean water to drink...and other people are breeding them for meat and leather. The point is, you can't control what other people do, and other people are evil, deranged, mentally ill, etc.
One solution to the dillema of what to do with the last two dogs or cats may be to just banish them to somewhere in the wild they can survive to start a wild line that may evolve into another species, like the dingo arguably is.