Much of the currently used techniques of scrubbing the carbon dioxide from power plant exhaust uses sophisticated amine and other organic compounds. These compounds are then forced to release the CO2 with electric current or heat so that the concentrated CO2 can be pumped underground or otherwise sequestered. Why not just spray the power plant exhaust with a finely atomized Sodium Hydroxide mist to directly precipitate out the CO2 as Sodium Carbonate or Sodium BiCarbonate? The carbonates can be sold as carbon credits and/or be useful products. Sodium Hydroxide can be obtained in quantity cheaply from electrolysis of sea water and other methods.
TL;DR we use stupendous amounts of fossile fuels, capturing the carbon would require even more stupendous amounts of sodium hydroxide.
The advantage of amines is that removing CO2 from amines only requires heat input and the amines can be recycled (though some degradation does occur).
You describe a caustic scrubber, chemistry as outlined here. you would need about 2 mol NaOH / 1 mol CO2 or about 6 kg NaOH per kg Carbon in the fuel. So imagine a coal plant, and for every train car full of coal six train cars of NaOH prills arrive.
This paper (that I only skimmed) estimates the carbon footprint of NaOh tp be 1.9 kg CO2 equivalents (per kg). Caustic scrubbing will not provide a net carbon benefit.
It is possible that someone, somewhere built a bicarbonate factory using exhaust gas as a CO2 source. The driving consideration re. sizing this plant will be: How much bicarbonate can I sell, not how much electricit do I need => how much coal do i burn etc.
I would guess it's because the amines are able to be regenerated, so they don't need to buy more that often. Where as with the sodium hydroxide, it would not be regenerated because the CO2 took it's oxygen. Basically, the amines do not react with the CO2, but rather they absorb it and then are sent to a desorber where the amines and CO2 are separated and the pure amines get reused.