Nature already has powerful carbon-removing technology, it's called a tree. Why reinvent the wheel? Simply fund reforestation efforts. Plus many wildlife species will be saved from extinction. For example, orangutans in forests of Borneo island. EDIT And besides, humans will survive climate change, we are 7.8 billion. But we made 1000s of species go extinct. So I say carbon capture is the wrong problem to solve.
A while back I saw a paper in Nature claiming that we could use trees to absorb up to 1 gigaton carbon annually, but more than that and we'd seriously impact biodiversity. There are probably other estimates around, but regardless of the exact threshold, it wouldn't be a bad thing to have technology able to do the same thing with a thousand times less land area. Then we can let the forests grow undisturbed.
To your edit: climate change itself is making species go extinct, so it's not like we can solve the extinction problem without addressing climate change too.