"Talk American" is an episode of the podcast Code Switch. In it they note the role of Hiram College professor John S. Kenyon of creating the name, and argument, that the "General American" accent was the accent of Cleveland in that era.
This sparked an idea that I have no way to prove. Cleveland is an odd part of the Midwest. It was originally claimed by Massachusetts as its "western reserve" (preserved in the name of Case Western Reserve University), and was long thought to have a more New England feel than Columbus or Cincinnati, Ohio's other big cities.
It makes sense to me that in the 1920s, Boston was long past the time when the evolved accent of the Puritans still dominated ordinary speech. The mass Irish migrations of the 1840s, in particular, would likely have dramatically changed the speech of "old" New England.
Cleveland on the other hand, might still have had a Puritan-derived speech among its dominant class, even as the wave of Eastern Europeans was arriving at the bottom of the class structure.
The claim for the General American accent today is that it is the standard broadcasting language, used by news readers to bring the "news from nowhere" without an identifiable accented location.
Puritan culture has always had a claim to set the standard for American high culture. It makes sense to me that its linguistic descendant is still the closest thing we have to a standard way of speaking.