The storage part is a big issue with current plans for CCS and DAC. If they end up turning around and repackaging the CO2 into something that can be released again, it's only better than pumping out new petroleum and not a true reduction.
But the title is a bit wrong. It should be "Pulling CO2 out of the air alone won't stop climate change". Even if we dropped the CO2 level to preexisting levels (280ppm or even lower) there are still some feedbacks already in play that wouldn't suddenly stop their contribution. It would be far better than continuing to add to CO2 for sure, but there is still going to be some climate change and adaptation needs regardless of how well we act. And of course, we are not acting very well.