You can make dry ice directly from air, by cooling and compressing it, because after the condensation of water the other gases have much lower boiling temperatures than carbon dioxide.
Obviously, the efficiency of the cycle is lower when starting from air instead of from pure carbon dioxide, but the input energy is unlimited, as it comes from solar or wind energy.
This does not compete directly with batteries, which have very good cycle efficiency but poor mass, volume and maximum storage time. It competes only with other energy storage technologies that can also store energy for many years and/or which have comparable energy per mass and energy per volume ratios.
To judge whether the technology of this company can be competitive, we need some numbers that they do not provide. Nevertheless, it is also impossible to dismiss it a priori, because there are good chances for it to have a good enough efficiency.
What they say about having a better efficiency than the 2-step process through carbon monoxide is plausible, and even the indirect Fischer-Tropsch synthesis has been economically viable in the past, in the absence of enough fossil fuels, despite the fact that it is not competitive today, as long as the fossil fuels are still cheaper.