it's actually much easier than this makes out. Sodastream bottle, some kind of cloth like a tea towel, pillowcase or even a sock over the end, let 'er rip and you have a chunk of dry ice.
At that point, why not just buy the dry ice directly? It would surely be cheaper...
But, based on the write-up here, it sounds like he was doing this out of curiosity, at which point yes buying the CO2 would sort of defeat the point.
Of course, one could also ask the same questions about buying the baking soda and acid. There is no clear delineation at which point before which it's "made from scratch" vs. "store bought".
Sounds like the answer in this case was: the person in question thought that starting from purchased baking soda and acid was an interesting problem to solve, and starting from bottled CO2 was a boring problem to solve and starting from some precursor to baking soda and acid was too much of a pain to solve.