So they have to procure an amine in the pure form,
mix it with purified water,
heat it to 125 to 165 oC (a lot of energy, also under pressure),
Partially Correct - there was no mention of it being under pressure.
bubble the air through it (requiring at least the same pressure as the solution so there wouldn't be backflow)
1.001 Atmosphere of pressure, yes.
then recover the product using distillation (energy intensive).
Correct. Though as the solution was originally at 125-165C, the methanol would most likely be in vapour form, so condensation would be a relatively simple matter.
It's good chemistry and interesting catalysis, but I don't see how it will be cost-effective.
My guess is it would be cheaper to let a tree reduce the CO2, chop it down, and make the wood alcohol from that.
Oh, of course. Though that is a process that requires considerably more time than this one, it seems.