I get that from a close reading of Rutherford's own pronouncements in his various major speeches of the time, as well as the overall thrust of the content in "Consolation" between 1937 and 1941. The constant attacks on "big business, the clergy, and politics" heard at the time would have been very familiar and would have appealed very much to the same audiences who were reading pamphlets and books by such figures as Earl Browder, John Spivak, and George Seldes, all of whom had strong credentials on the left. Both Spivak and Seldes wrote extensively on the Catholic church's role in right-wing/fascist movements in the US during the middle and late 1930s, and both were quoted with approval in Consolation at various times.
That's not to say that the JWs followed the CPUSA line, or anything that specific -- although you'll find some interesting positive mentions of the Soviet Union showing up in Consolation from time to time, even as the Judge was careful to condemn "Communism and Naziism" in the same breath -- but I think, from my own readings of Communist and Socialist publications of the period that a "Daily Worker" reader could easily have tuned in on the broadcast of, say, "Fascism or Freedom" and would have nodded in agreement at frequent points thru the speech. And although there's a strong thread of isolationist thought in Consolation once the war broke out in Europe, it's important to note that the CP line in 1940-41 also opposed US involvement in the war -- and was thus as much a "leftist" position as "rightist" one.
Now, there were certainly points of convergence between true right-wing movements like that of Father Coughlin (his vision of "Social Justice" was essentially constructed on the Fascist-corporate state model) and some of the things that came out of the Judge's mouth, especially in his condemnations of FDR -- which I think stemmed more from his unsatisfactory dealings with the Roosevelt appointees on the FRC/FCC in 1934-35 than any thing else -- but the substance of Rutherford's message strikes me as rather militantly and explicitly anti-capitalist at its core, and that tracks quite closely with the main thrust of 1930s left-radical thought boiling around the working-class activists of Brooklyn and Detroit. For that matter, the CP itself was strongly anti-FDR up until the emergence of the Popular Front period in 1936.
Incidentally, I had no idea until recently that H. H. Stroup lived until 2011 -- and he lived his last years not far from me. I'd have enjoyed a conversation with him. (And while he was raised a Presbyterian, he became, pastored, and died a Congregationalist, a denomination which we New Englanders sometimes call "dishwater Christians.")