Yes - it might have impurities but we can store them along with the CO2.  IMHO the only feasible way to get the CO2 out the atmosphere is to bio-engineer stuff that deals with it in the oceans.  Some hybrid that photosynthesizes like mad.  Yes,  its a desperate measure.

For the demand side:  What other approaches other than turning off the engines?

You don't really need any bio-engineering, just some plain (and quite simple) engineering. There is an idea of using iron fertilization to increase phytoplankton growth in oceans, which could potentially make a big difference. There are lots of uncertainties, tough, like what it would do the ecosystem.
There was an old lady that swallowed a fly... We'd want to be desperate before going too far with this, no?
No, all you have to do to get CO2 out of the atmosphere is to grow biofuels and not dump the carbon back.  I proposed a model in Going negative.