This sort of operation would need a stable cap and trade or carbon tax system to be profitable.
I should also add, I think China or somewhere in the hot parts of Africa would be a a good place for this as it would be a large area operation. Sadly as an Australian we suffer from regulatory capture and a carbon tax won't be back on the cards for some time.
IMO, barring some breakthrough, biology will be the most efficient mechanism for capture. Millions of years of evolution have made plants and algae pretty good at carbon capture as the carbon levels slowly depleted (until recently).
An appealing benefit of biological carbon capture is the fact that all biology is solar powered. All the non-biological solutions required a pretty hefty energy input from somewhere. I mean, I guess it's nice that you could provide that from solar/wind/or nuclear. But then it's also nice to not require transmission lines for energy input.
I'm hoping someone can find, or mix and match an algae in lab that will have the right mix ofproperties to explosively grow.
Providing food stock will be tricky, either through something very efficient like straight sugar or designing the algae to consume anything which could have other upside benefits.
And like all well meaning engineering the real challenge will be making sure the algae that can grow by eating anything doesn't get out of the pond.
But if you provide too much nutrient, too much algae grow, and when they die and decompose they absorb all the oxygen in the water and kill the whole ecosystem. (That's called eutrophication).
Charcoal is pretty much pure carbon and that's what we should be putting into the ground. Putting CO2 into the ground runs into problems because you have to deal with the fact that you are burying a gas (gasses don't like to be contained).
So skip that, and instead focus on burying pure carbon.