Is there proof that Jerry Siegel and Joe Shuster's Superman was directly inspired by Friedrich Nietzsche's Übermensch?

Question

Jerry Siegel and Joe Shuster were born in 1914, met in 1931, and in 1933 produced their first published work using the name Superman in The Reign of the Superman. It was a lightly-illustrated short-story featuring the titular character as a telepathic villain. The influence of Nietzsche on current events and Siegel and Shuster's Jewish heritage often get conflated by commentators as directly causal. Their 1933 Superman allegedly a repudiation of the adoption of Nietzsche's Übermensch, casting it as a villain instead of the over-man savior.

Yet this common narrative seems inherently flawed. If Reign was a repudiation of a philosophical influence by making it a villain, why would they make Superman the hero proper in Action Comics #1 published in 1938? After Hitler was more clearly problematic to Americans?

Siegel and Shuster were 17 when writing Reign and it seems a leap that they would be so conscious of current affairs and geopolitics to: single out a politician a world away, know his philosophical influences, and write a story specifically meant as critique... when mainstream American press had yet to take Hitler wholly seriously yet and whose persecutions began in 1933, after Reign had been written in 1932 and published in January.

Meanwhile, the spread of the term "Superman" during Siegel and Shuster's teens in pop culture and pulp magazines is well documented. Houdini, Tarzan, Doc Savage, and a litany of other literary, scifi, and genre heroes were called and advertised as "Superman" at the height of their teenage content consumption. On various occasions they would cite these as among their influences for Superman. (Example: Interview with Siegel, Shuster, and Joanne Siegel conducted by Tom Andrae, Geoffry Blum, and Gary Coddington, NEMO: The Classic Comics Library, issue #2, August 1983)

Question: Are there any original non-hearsay sources proving 17-year-old Siegel and Shuster created Superman directly in response to Friedrich Nietzsche's Übermensch?

Every source on that matter I can find seems to be the author of the article leaping to the conclusion rather than sourcing Jerry or Joe.

Answer 1

Chapter 43 of Nietzsche's Thus Spake Zarathustra contains:

I long to be up, out, and away to the Superman!
— Wikisource

which is remarkably similar to the comic character's "up, up, and away!" catch phrase.

It's not solid proof, but it's a good indication that Siegel was familiar with the work.

Answer 2

The Agatha Christie short story, The Manhood of Edward Robinson, ends with the line, “almost as Bill the superman might have done”. https://books.google.co.uk/books?id=3jbKEAAAQBAJ&q=Bill#v=snippet&q=Bill&f=false

The story was first published in 1924, then again in June 1934 as part of the collection The Listerdale Mystery

If ‘S&S were voracious consumers of pop-culture’ it seems likely they would have read Christie’s work, and could have been influenced by the line.

Alternatively it could be seen as evidence that the concept of a perfect ‘superman’ was sufficiently common coinage in the ‘20s that Christie could use it as a tangential archetype and assume her readers would understand.