What is the known cheapest chemical solution to extract CO2 from air on industrial scale?
Your intuition is correct, and such compounds are plenty. The cheapest are probably $\ce{NaOH}$, $\ce{Ca(OH)_2}$ and the like, but these are not easily regenerable. Then there are organic amines, like monoethanolamine, which can do the same in a reversible manner. Indeed, they are used in industry.