When I think about carbon extraction and storage, I get the same feeling as when I read about the ancients discovering oil seeps and surface deposits of hydrocarbons. They knew that they could be harnessed for keeping fires lit and even sometimes associated them with the supernatural. They just lacked the tech and industrial background to use them. We know that carbon is essential for life and if we ever engineer use…
What is hard to find - and has always been hard to find is cheap energy. And carbon dioxide is essentially the lowest energy state for carbon. It's a dead end as far as energy goes. That's why it's a waste product.
If you want to build useful structures then you're still subject to thermodynamics, whether you're using "organisms" to do it or not. So using graphite or any other form of carbon would be a better source of carbon than CO2. The only reason to use CO2 is that it is commonly available (which is why plants use it).
And again, no one is proposing removing all the CO2, just enough to keep us in this little ice age we have been enjoying. So there's no reason to worry about running out.