You're mostly right, in that a timing issue could be resolved through a temporary storage unit, though probably wouldn't have to put it underground.

Likewise, batteries/capacitors/whatever would probably be set up on the electricity. A plant design would seem to want to control the flow rates of both.

Overall, the parent comment's unlikely to be correct in its speculation that the reason for open-air capture is related to timing issues.

You'd have to separate pump and store it, when electricity is expensive.

I suspect the advantage of this system is that is doesn't need pure co2 to work

And if you were storing CO2 I suspect the first question is how do we liquify it. So the advantage fades away.

edit: You'd probably want to store the electricity on-site instead for this. At least if there's no CO2 to consume you can be in the electricity storage business.