A very interesting review in this week's New York Review of Books (I don't have the article or the reviewer's name to hand) of the whole series. I cannot comment on any of this, having only seen the first in the series! Just reporting!
Apart from the usual cricitisms, the author argues:
1) That the America, America stuff is overdone, and points to the lack of reference to Latin American music throughout, including Tizol and other important people;
2) The portrayal of Duke Ellington recycles the sole hero model, underplaying the work of Strayhorn, but also that Ellington was a consummate arranger and re-arranger of the ideas of his band members;
3) He is bothered by what he calls very subtle intimations of racism and anti-Semitism in the series. These include the casting of the blacks as intuitive musicians (as opposed to expert musicians, craftsmen, etc.); and the references to the two main Jews in the series Benny Goodman and Artie Shaw as in it for "the fame and fortune", obliquely focussing on their desire for money, etc.

yours, Peter T.