Scientists say that we’re in the midst of a mass extinction due to climate change, overfishing, plastic pollution, etc. If/when species go extinct, will nature not just reorganize itself and produce a new, healthy biosphere? Hasn’t earth been doing that for billions of years?

Eventually yes, but not at the rate theyre currently going extinct. That's why its a mass extinction.

Eventually it will. But certain animals going extinct can destroy a large chain of animals that are dependent on them

Eventually yes, in theory, if things don't get so bad that no life can live here

But a key difference between this and other extinction events is that this is human made.

Maybe. Each extinction has knock-on effects. What they will be is hard to predict.

Eventually yes, but that is really bad for humanity, because the "eventually" could be a million years later, and humanity wouldn't have that kind of time if a lot of critical species went extinct.

What is your point?

That we should be ok with terraforming earth into a barren husk with just us and the cockroaches because "new things will evolve eventually"?

Something has to survive to fill the ecological niche.

The Anthropocene Era is simply changing too rapidly for adaptation to be a viable genetic strategy. Extinctions are happening much more rapidly than new ones are emerging. It's that rate that is the problem. As long as extinctions outnumber new species, we are reducing the overall size of the ecosystem, and that can lead to an ecosystem collapse.

Why do you figure that "us" won't be one of the extinct species?

There are enough billionaires with bunkers that I am not really worried about humanity going extinct.

Whether it is a world worth living in is another question.

Yes, but not within our children's-children's-children's lifetimes. Life on earth will go on but recovery millions of years from now won't do anything to help us now when we live on a devastated planet that's a shadow of its former self.

This is the thing. We're not trying to "save the planet", long term the planet will be fine.

What we're trying to do is avoid ruining the place where we live. If we ruin it; on a geological timescale it will be fine, but on a human timescale it will be ruined