Surely the same fallacy is the reason we care about these species in the first place?

You can kind of make an argument from entropy, an extinction cuts off our access to information about a species. But if that were the reason, taking samples for DNA and then killing off the animals would be okay. We care about extinction because we like nature and want to preserve it.

It cuts both ways. Having more species is not inherently better.

Climate change is a battle against ourselves, for ourselves. We will be the ones to pay the cost (potentially with our lives, or way of life at the very least); nature will persevere whether we win or lose.

Life will always find a way no matter how much humans (or ice ages, or asteroids, or whatever) screw up this planet.