I'm not entirely sure if you do have a misunderstanding, which is why I asked the question. I'm trying to figure out why you think "carbon capturing" is equivalent to "being able to re-mine hydrocarbons"?
Assuming for the moment that the rest of the process is perfect (that is: ignoring that it costs energy to perform carbon capture, ignoring that it costs energy to convert from captured carbon back into some sort of usable hydrocarbon, etc.):
it is already very challenging to get to even 50% efficiency in the engines involved in converting a fuel into electricity; whether that engine is an ICE, or a steam engine often used to convert certain renewable sources of energy into other forms.
Most of that lost efficiency is due to fundamental constraints of thermodynamics : https://en.wikipedia.org/wiki/Carnot_heat_engine
(Essentially, useful engines that produce mechanical work from difference in average energy between two bodies operate in cycles. Suppose we start at time 0, and it takes k time ticks for a cycle to start from its initial state X(0), go through its processes, and return back to the initial state: X(k). The fact is that X(0) and X(k) must be different, they cannot be exactly the same, because then that would mean that it is impossible to distinguish between X(0) and X(k), and your engine is not just an engine, but a time machine. (This is why entropy is closely related to the fact that time is an "arrow": always moving in a particular direction.))
All this to say: it is not possible to use a finite source of energy for a "long time", unless the finite source dwarfs by "many orders of magnitudes" (how large the magnitude, determines how much time before it runs out) the energy drained from it per time tick.
So even if we got carbon capture working as a way to "recycle fuel" (totally, totally ignoring the fact that it will cost more energy to do the carbon capture and store said captured carbon, than the mechanical work we get out of it (the nuclear fusion problem, except not even technically solvable)) it would not last us for more than a few seconds.
(Existing hydrocarbon reservoirs do dwarf our current energy use per time tick, but not by that much, and not if we also want to accommodate material growth, because of the energy cost of also "safely managing" the byproducts of the hydrocarbon->mechanical work process.
The sun in particular, massively, massively dwarfs our current use per time tick. This is what makes "renewables" renewable: they have a massive bank of energy banking them. We have an awe-inspiring fusion reactor just throwing energy at us for a while. How do we convert it into usable work?)
https://e360.yale.edu/features/three-myths-about-renewable-e...
Putting all that together as the context then, my question to you would be: how can one still use captured carbon as a sort of battery in any meaningful way? Is there a misunderstanding of thermodynamics involved on your part, or could you help me understand where my misunderstanding around thermodynamics lies (I am not an expert, just a novice)?