The reports of Johnson's recording being played at the 'From Spirituals To Swing' concert derive from an article written by John Hammond, published on Page 27 of the December 13th 1938 issue of New Masses, a pretty strongly leftist weekly magazine.

The mention of Johnson is a paragraph within a larger article written about organising 'From Spirituals To Swing', and trying to contextualise what he was aiming for in the concert - a celebration of the talents of black musicians who usually are swept under the carpet ('the Negro musician still finds himself oppressed and ostracised. Out of the hundreds and hundreds of musicians employed by radio broadcasting studios in house bands not one is a Negro. With the exception of New York City...the AFL craft setup effectively isolates him in jim-crow unions, where he has no protection').

Around the middle of the article, there is a paragraph devoted to Johnson:

Robert Johnson was going to be the big surprise of the evening for this audience at Carnegie Hall. I knew him only from his Vocalion blues records and from the tall, exciting tales the recording engineers and supervisors used to bring about him from the improvised studios in Dallas and San Antonio. I don't believe that Johnson had ever worked as a professional musician anywhere, and it still knocks me over when I think of how lucky it is a that a talent like his ever found its way to phonograph records. At the concert we will have to be content with playing two of his records, the old 'Walkin' Blues' and the new, unreleased 'Preachin' Blues', because Robert Johnson died last week at the precise moment when Vocalion scouts finally reached him and told him that he was booked to appear at Carnegie Hall on December 23. He was in his middle twenties and nobody seems to know what caused his death.

In his place we are bringing in from Chicago a fine blues singer who goes under the name of Big Bill, who used to make records for Perfect and can now be found as the best seller on the Vocalion 'race' series. He also plays the guitar and accompanies himself.

Hammond may have changed his mind in between writing the piece and the day of the concert, of course, but as the organiser of the concert he was probably in a good position to know what the organiser of the concert was organising re: playing a phonograph recording of Johnson. That Johnson merits a paragraph or two in the article, as far as Hammond is concerned, is probably a pretty good indication that he did indeed play the phonograph recording of Johnson. To the extent that the article was an advertisement for the concert, he was trying to advertise the blues as being fascinating because it is 'instinctual', because Johnson wasn't a professional (despite Hammond's prose, he certainly was a professional).

Hammond also makes a big deal in the New Masses article about how the talent available in 1937 was only satisfactory compared to the wealth available even a few years ago. His discussion of Johnson in the article comes directly after his discussion of what has been lost before 1937. The recent and excellent book by Conforth and Wardlow about Johnson's life, Up Jumped The Devil, portrays Hammond's choice of recording as also going along with Hammond's frame for the blues at the concert; 'Preachin' Blues' and 'Walkin' Blues' are amongst Johnson's most 'primitive' or 'simple' blues, and Hammond of course was trying to portray a progression in the music from primitive to sophisticated in the concert as a whole. So you can see why, with what he was aiming to do, Hammond might play recordings by a dead blues guy: he is trying to make the big statement of 'this is a pretty pure form of the blues at the heart of African American music, and we have already lost it'.