Well, Strand was first influenced most by Lewis Hine, who was his mentor. But it was Stieglitz who inspired Strand to pursue a career in fine art photography and gave him his first show at the 291 Gallery in New York. I'd call that pretty heavy influence. Of course, 1927 is somewhat early in an absolute sense, but not as far as fine art photography is concerned. By then, 291 Gallery had already been closed for a decade. Paul Strand, Edward Weston, Edward Steichen and Imogen Cunningham, to name a few, were well established artists. Atget became a major influence much later in the century--his influence on early 20th century photographers was minimal. It wasn't until the Museum of Modern Art purchased the Atget collection in 1968 that his work became widely known and he was given the recognition he richly deserved.
Pop, I guess that's one way to look at it, but a number of the people you mentioned were aware of Atget's work early on through Abbott or through association with one or another of the artists who bought his "documents." HCB certainly was. In The Decisive Moment
he says, "... I met photographers who had some of Atget's prints. These I considered remarkable and, accordingly, I bought myself a tripod, a black cloth, and a polished walnut camera three by four inches..." This, of course, was before he turned to the Leica.
In any case, I think Stieglitz was a fine showman but not a great photographer. People seem to be confused about his photography because of the central role he played in making the work of some of the world's finest artists available to audiences in the U.S. Other than a few pictures such as The Steerage, The Terminal, and my personal favorite, Fifth Avenue, Winter, what great photographs did he produce? Yes, he got Strand to do serious photography, which was a plus, but later he turned up his nose at Walker Evans's work, which, in my estimation, removes a lot of the gilding from his crown. Seems to me an awful lot of Stieglitz's glamor came from his affair and later marriage with Georgia O'Keeffe. Actually, I see Stieglitz sort of as the Barnum of the fine art field.