unless those species are dying because of human activity, why would we want to save them?

Because elephants and rhinos are cool and interesting, and their existence doesn't cause substantial harm to us or the planet. Do we really need another reason?

I don’t think there’s an ecosystem left that is free from influence from human activity so you would have to include all of them for that reason alone. Preserving genetic diversity is probably a good enough reason, you never know when that next asteroid is due.

genetic diversity for its own sake is hard to justify as a goal, or else indeed we wouldnt be fighting this virus to extinction

It's really easy to justify. A very prominent example of why would be penicillin. Who knows how many useful things we'll find and where?

"Unless"? Yes that is exactly why they are threatened. Have you been living under a rock?