They attribute it to the original mistaken fixed-price bottling contract, Coke's response of pushing the price down with "predatory" marketing of a 5c price point, and the rise of vending machines and their inability to make change until the late 40's.
The best part of this story, and I'll just go ahead and spoil it for you: late in this drama, Coke decided they wanted to raise prices, but since vending machines couldn't reliably make change they were stuck with a nickel-denominated price and couldn't double it. So instead, they stocked vending machines with "official blanks"; one in every 8 times you went to a Coke machine for a drink, you got an empty bottle, and had to pay again, which raised the effective price of Coke across all consumers by less than a cent.
In the podcast they say that idea "never saw the light of day".
It's still the best part of the story. Followed by the request for a 7.5c coin.
A less informed reader may be unaware of this and be reading the article from the context of modern inflation, for which a 70 year history of no price change would be much more remarkable.
I am not sure what you mean here - "of more questionable accuracy [than CPI measurements today]", or "of more questionable accuracy [than gold in reflecting the cost of producing a coke]"? If the former, I grant it, but not (automatically) that it's enough to make a difference. The latter seems absurd - as I elaborate on a bit below.
"CPI measures many different types of goods (not all go up together)"
Coke is made out of many different types of goods (in terms of everything it takes to put a bottle on a store shelf), plus labor of people who want to buy many different types of goods. CPI should be expected to track the cost of producing a coke quite a bit better than a single commodity that isn't even involved in the manufacture and which experienced changes in supply that had nothing to do with coke.
"[A] doubling of CPI over that many years is relatively low inflation."
Relatively low, but not absurdly low, and certainly not completely flat; and note (!) that the end points picked here are not representative. 1930 CPI is the start of the depression; inflation in the 1920s was higher than anything we've seen since (says wolfram alpha) and the price of coke did not move.
To the general, implied counterfactual - "would Coke have raised it's prices sooner if there were much higher inflation?" - I think we have to answer "of course", but that doesn't necessarily mean that unusually low inflation was a significant part of it. Earlier periods have experienced less inflation, but coke's price lasted an unusually long time.