Thus my debatable but possible observation that Mr Hawkins and his colleagues eventually gave that strange horn a long awaited recognition.
I think you're right - debatable.
Certainly Sousa and Gilmore, and the thousands of town and mill bands that emulated them, found a place for the saxophone - a place closely related to its original purpose which was to provide a loud reliable bass instrument for military bands.
Then we have the military bands of the World War One period, many of which had saxophones in them.
After WW1, then, there existed a number of people who had some degree of experience at playing saxophone, and there was some amount of inventory of saxophones, in pawn shops, military surplus, etc. (The US military shrank VERY QUICKLY after the Armistice.) It was logical, then, that saxophones would be used in popular dance music as well.
I believe that Jimmy Dorsey, Frankie Trumbauer, and Rudy Weidoeft pre-dated Coleman Hawkins in public visibility. Quite possibly, Sidney Bechet as well. On the other hand, there's absolutely no doubt that Hawkins established one of the two major schools of jazz tenor saxophone playing and was enormously influential up to the present day. So if Coleman had decided to stay with the cello, how would the history of the saxophone have been different? You can see in those old band photos from the 20s that the forest of woodwinds in front of the band was a standard feature of instrumentation. It might have been that no single commanding figure would have emerged in jazz saxophone the way Hawkins did - but that might simply have meant that it would remain more like piano or trombone in the 20s and 30s, still fully integrated members of the jazz ensemble, but without that single dominating figure.
I suspect that the rise of the saxophone in popular music and jazz in the late 20s on into the decades following would have happened anyway - the cadre of musicians who knew how to play it, the availability of instruments, and the tonal flexibility and ability to project in a dance hall were all there.