I'm not so sure it was playing in marching bands that gave him that power. Other New Orleans clarinetists who were more noted for local band work, like Alphonse Picou or Big Eye louis Nelson, were not so dominant. Bechet left NO by the time he was 20 but no doubt he absorbed much from the general musical atmosphere and from hearing and playing with other strong players - trumpeter Freddie Keppard comes to mind. Bechet was also an educated creole and was taken to the ballet as a boy. This may have added to his sense of drama. (He wrote his own ballet in the 50's.). He was also taught by some of the best NO clarinet teachers - Lorenzo Tio and George Baquet.
I'm inclined to think though that it was this mix plus his own dominant personality that enabled him to be so powerful and communicate so directly. He was certainly pugnacious in his young days - deported from UK (despite playing for Royalty), jailed in Paris, trying to start dog fights with other musicians' pets in Germany - yet in later life interviews his voice and demeanor is soft and mellifluous.
Above all whatever people think of him now, he was the first jazz musician to explore the soprano and no-one else (save Johnny Hodges, whom he taught) has ever sounded like that. His heavy vibrato is criticised now (why did it get wider and deeper as he got older? "senility, dear boy" he told young reporter) but it was a feature of NO orleans playing and much admired then. I doubt that without the foundations he laid for soprano that the likes of Coltrane, Shorter, Garbarek (who in turn was influenced by Coltrane) would have easily found a voice on the instrument.
The British trumpeter and writer Digby Fairweather sums him up well "His compositions were melodically stronger than those of any other classic jazz musician. his creation of a vocabulary for his instrument was as great as Coleman Hawkins' for tenor, and his records are majestic".
