Louis Armstrong was influential on music, but much less so on American society as a whole. I could probably name a couple dozen American musicians and composers who were far more influential than Armstrong. Just because someone comes along earlier in chronological time doesn't mean that they are more influential on society than someone who came later. When I think of influential musicians, I think of people who not simply successful musicians, but who also fundamentally transformed the way men and women, different races, and different generations interacted with one another. George M Cohan probably was second to Elvis in affecting society. His music altered the psyche of the American people and how they perceived themselves to be Americans.
>I could probably name a couple dozen American musicians and composers who were far more influential than Armstrong.<
I couldn't disagree more. Every popular singer after Armstrong owes a debt to his style and his phrasing -- whether they know it or not. His influence has been significant even in country and western circles, having been transmitted there by the brand of jazz called "western swing." (Just ask Willie Nelson!)
And ditto for every instumentalist since the 1920's (and right up until today) who has played an improvised solo -- not only in jazz, but also in blues, rock, straight-ahead C & W, and even bluegrass. That's because Armstrong virtually invented the improvised solo as we now know it.
Of course, we of the present era take such matters for granted. The situation calls to mind an old story about a college student who upon leaving her English class asked, "What was so great about Shakespeare? All he did was string together a bunch of clichés."