Issue is that batteries are getting so cheap its going to be better just to use them in almost all cases (aside from niche stuff like making rocket fuel on mars).
It's not just cheap that's useful (and, as per recent "mineral grab" politicing, that's only so long as the lithium/whatever keeps flowing in whatever amounts are needed to
make(/recondition) the current required stock of batteries).
Batteries are heavy, for the energy they contain, even using the aforementioned lithium (the lightest you can go, with that particular choice of battery chemistry). And they stay the same weight (or, in air-breathing hydrogen cell technology, get
heavier) as they are expended. A big problem for pure-electric planes. And would be for rockets (assuming that a pure-electric rocket was possible). There's at least
one current rocket company (ULA? RocketLab? I forget, but... Yeah, probably the NZ-based one) that actually use batteries to power the rocket turbo, and
specifically have a staging point at which the jettison the depleted (first) battery to save crucial weight, even as they continue to thrust with their actual rocket fuel. Energy density is a better equation for burnable fuel (even fuel+oxidiser, or hypergolic all-in-one), and the casual disposal of the battery is an unusual (but necesary?) solution, compared to practically all other rockets which tap off (in open-cycle) or into (closed-cycle) the existing rocket-fuel supply to power any pumps needed to burn through the 'main' fuel+oxidiser. The fuel solution also automatically makes itself lighter, as there is less of it left, before considering the disposal (or recovery) of the engine, tank and rest of its structure at each stage of staging.
I know I equated it to a battery, taking the carbon out of the air (as you say, and I intended to, merely temporarily), but it is in some ways better. You need huge batteries of... erm... batteries[1] to keep the same amount of photovoltaic energy 'hanging around' for future use, compared with relatively smaller vessels of photo-derived hydrocarbon. (Might depend what form you want to use it in, later... back-converting manufactured methane to electric to charge your Tesla
might be less efficient than doing some sort of gas-turbine thing, Mad Max-wise, with your off-grid petrol-free/mains-free automobile. And hydrocarbons can be polymerised into useful physical materials for long-term use that stored electricity cannot, by itself.)
It's all a balance, and using small amounts of power over a long time (and/or massively in parallel) to produce burnable fuel that can send a plane into the sky (or a rocket to Mars; and, perhaps more importantly, a rocket
returning from Mars, when the only other known fuel practically available there is that which we take from Earth) is perhaps the most efficient way of give the energy needed for the more concentrated Ooomph of any aerial vehicle (though cars, lorries, boats, subs, etc may be more usefully fully-electrified). They
are working on electric aircraft, I know. Hobby-drones up to microlights are now fairly trivial possible battery-users, light-planes are getting experimental about it (and using the advantages of many smaller motors to work with flexible VTOL solutions that were difficult with non-electric power sources), but getting into the realms of anything approaching ocean-/world-spanning mass transportation is going to be awkward. I could see a market for creating an AvGas-compatible air-sourced fuel, temporarily soaking up the equivalent carbon dioxide from the air (and, hopefully, a large quantity of off-peak renewable energy) as the most immediate way forward. It'll borrow from the wind (physically
and energetically) and then get dissipated back again, entropic losses as (almost) the only net pollutant.
Though the necessary infrastructure to support this still means building windcturbines, etc, which are (currently) reliant upon resource extraction. Maybe if the carbon from decarbonised air can be sequestered into fairly stable chunks of carbon-fibre/etc (perhaps a better use, but harder, than pumping liquid/gas "manufactured hydrocarbons" back down the very wells that we took so much of the "natural hydrocarbons" out of) then it'll not just be (barely!) making the anthropogenic imbalance of natural carbon-balancing worse.
Not that I'm approaching this from any full-blown "just stop oil" perspective, I'm just pondering the full cycle. All methods to take all-too-plentiful CO
2 out of the atmosphere (and I'm not sure which one that article is saying - sounds 'new', but no detail given and could just be a 'better implemented' old one) sound good, though creating it again later makes it only carbon-neutral (ignoring overheads, losses[2] and initial/continuing hardware manufacture). Until it produces every grade of hydrocarbon that we currently expect to use (based upon what hydrocarbons we got from the ground, so commercially strived to find a use for, everything from methane to the tars we create our roads with)
or lets us finally forget about needing them (is there some better road surfacing method?), someone will be sucking 'old' hydrocarbons out of the ground and (hopefully) supplying just enough for all remaining needs. "Air-source hydrocarbons" might be good for more and more temporary (create-to-burn) or more permanent (sequestered in building/item structures of all kinds, at least until the enclosed lithium battery gets damaged and sets the whole thing on fire) carbonaceous uses, as they become more attractive (financially, politically, philosophically, existentially, practically) than Old Oil. Time will tell where we get to, in this sort of approach.
But we do
also need the electrical batteries (or
equivalent power-sequesterers) to buffer the power needs against the power availability, for at least those demands that can't be defered (perhaps to act as power-storage/-supply by proxy, like small-scale solar-to-hydrocarbon-to-storage-to-generator-to-electricity units, as per demand). All very complicated, of course.

[1] Ultimately, of course, it's batteries of individual
cells, but terminology gets funny when plant cells and fuel cells are related things, which of course are also seldom singular in practice!
[2] Converting CO
2 (+ water) to CH
4 and then
losing it, into the atmosphere, sounds worse to me, without doing a full audit. But methane is more of a greenhouse gas, so I suppose it hangs on whether you're doing anything significant to the atmospheric water vapour, as well, enough to compensate... And more cloud cover (and snow) is what we want, not less, while ground water darkens the surface and makes it less reflective. So that's not just changes to the carbon cycle we're having to consider.